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HILBERT AND CHOW SCHEMES OF POINTS, SYMMETRIC
PRODUCTS AND DIVIDED POWERS
DAVID RYDH
Abstract. Let X be a quasi-projective S-scheme. We explain the relation
between the Hilbert scheme of d points on X, the dth symmetric product
of X, the scheme of divided powers of X of degree d and the Chow scheme
of zero-cycles of degree d on X with respect to a given projective embedding
X ,→ P(E). The last three schemes are shown to be universally homeomorphic with isomorphic residue fields and isomorphic in characteristic zero or
outside the degeneracy loci. In arbitrary characteristic, the Chow scheme coincides with the scheme of divided powers for a sufficiently ample projective
embedding.
Introduction
Let X be a quasi-projective S-scheme. The purpose of this article is to explain
the relation between
a) The Hilbert scheme of points Hilbd X/S) parameterizing zero-dimensional
subschemes of X of degree d.
b) The dth symmetric product Symd (X/S).
c) The scheme of divided powers Γd (X/S) of degree d.
d) The Chow scheme Chow0,d X ,→ P(E) parameterizing zero dimensional
cycles of degree d on X with a given projective embedding X ,→ P(E).
If X/S is not quasi-projective then none of these objects need exist as schemes
but the first three do exist in the category of algebraic spaces separated over S
[Ryd08, Ryd07b, I]. Classically, only the Chow variety of X ,→ P(E) is defined but
we will show that for zero-cycles there is a natural Chow scheme whose underlying
reduced scheme is the Chow variety.
There are canonical morphisms
Hilbd (X/S) → Symd (X/S) → Γd (X/S) → Chow0,d X ,→ P(E k )
where k ≥ 1 and X ,→ P(E k ) is the Veronese embedding. The last two of these
are universal homeomorphisms with trivial residue field extensions and are isomorphisms if S is a Q-scheme. If S is arbitrary and X/S is flat then the second
morphism is an isomorphism. For arbitrary X/S the third morphism is not an
2000 Mathematics Subject Classification. 14C05, 14L30.
Key words and phrases. Divided powers, symmetric powers, Chow scheme, Hilbert scheme,
Weighted projective spaces.
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2
DAVID RYDH
isomorphism. In fact, the Chow scheme may depend on the chosen embedding as
shown by Nagata [Nag55]. However, we will show that the third morphism is an
isomorphism for sufficiently large k. Finally, we show that all three morphisms are
isomorphisms outside the degeneracy locus.
The comparison between the last three schemes uses weighted projective structures. Given a projective embedding X ,→ P(E), there is an induced weighted projective structure on the symmetric product Symd (X). This follows from standard
invariant theory, using the Segre embedding (X/S)d ,→ P(E ⊗d ). In characteristic zero, this weighted projective structure on Symd (X) is actually projective, i.e.,
all generators have degree one. In positive characteristic, the weighted projective
structure is “almost projective”: the sheaf O(1) on Symd (X) is ample and generated by its global sections. This is a remarkable fact as for a general quotient, the
sheaf O(1) of the weighted projective structure is usually not even invertible.
Since O(1) is generated by its global sections, we obtain a projective morphism
Symd (X) → P(TSdOS (E)). In characteristic p, the ample sheaf O(1) is not always
very ample, and thus this morphism is usually not a closed immersion. It is however
a universal homeomorphism onto its image — the Chow variety. In summary, the
reason that the Chow variety depends on the choice of projective embedding is that
it is the image of an invariant object, the symmetric product, under a projective
morphism induced by an ample sheaf which is not always very ample.
Given a projective embedding X ,→ P(E), the scheme of divided powers Γd (X)
has a similar weighted projective structure with similar properties. We define the
Chow scheme to be the image of the analogous morphism Γd (X) → P(ΓdOS (E)).
For a sufficiently ample projective embedding, e.g., take the Veronese embedding
X ,→ P(Sk (E)) for a sufficiently large k, this morphism is a closed immersion. In
particular, it follows that Γd (X) is projective.
Contents
Introduction
1. The algebra of divided powers and symmetric tensors
1.1. Divided powers and symmetric tensors
1.2. When is ΓdA (M ) generated by γ d (M )?
1.3. Generators of the ring of divided powers
2. Weighted projective schemes and quotients by finite groups
2.1. Remarks on projectivity
2.2. Weighted projective schemes
2.3. Quotients of projective schemes by finite groups
2.4. Finite quotients, base change and closed subschemes
3. The parameter spaces
3.1. The symmetric product
3.2. The scheme of divided powers
3.3. The Chow scheme
4. The relations between the parameter spaces
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HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
4.1. The Sym-Gamma morphism
4.2. The Gamma-Chow morphism
4.3. The Hilb-Sym morphism
5. Outside the degeneracy locus
5.1. Families of cycles
5.2. Non-degenerated families
5.3. Non-degenerated symmetric tensors and divided powers
References
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1. The algebra of divided powers and symmetric tensors
We begin this section by briefly recalling the definition of the algebra of divided
powers ΓA (M ) and the multiplicative structure of ΓdA (B). We then give a sufficient
and necessary condition for ΓdA (M ) to be generated by γ d (M ). This generalizes
the sufficiency condition given by Ferrand [Fer98, Lem. 2.3.1]. The condition is
essentially that every residue field of A should have at least d elements. Similar
conditions on the residue fields reappear in Sections 3.1 and 5.2. Finally, we recall
some explicit degree bounds on the generators of ΓdA (A[x1 , x2 , . . . , xr ]).
1.1. Divided powers and symmetric tensors. This section is a quick review
of the results needed from [Rob63, Rob80]. Also see [Fer98, I].
Notation (1.1.1). Let A be a ring and M an A-algebra. We denote the dth tensor
product of M over A by TdA (M ). We have an action of the symmetric group Sd
on TdA (M ) permuting the factors. The invariant ring of this action is the algebra
of symmetric tensors which we denote by TSdA (M ). By TA (M ) and TSA (M ) we
L
L
denote the graded A-modules d≥0 TdA (M ) and d≥0 TSdA (M ) respectively.
(1.1.2) Let A be a ring and let M be an A-module. Then there
L exists a graded Aalgebra, the algebra of divided powers, denoted ΓA (M ) = d≥0 ΓdA (M ) equipped
with maps γ d : M → ΓdA (M ) such that, denoting the multiplication with × as
in [Fer98], we have that for every x, y ∈ M , a ∈ A and d, e ∈ N
(1.1.2.1)
Γ0A (M ) = A, γ 0 (x) = 1, Γ1A (M ) = M, and γ 1 (x) = x
γ d (ax) = ad γ d (x)
P
γ d (x + y) = d1 +d2 =d γ d1 (x) × γ d2 (y)
d + e d+e
γ d (x) × γ e (x) =
γ
(x)
d
Using (1.1.2.1) we identify A with Γ0A (M ) and M with Γ1A (M ). If (xα )α∈I is a set
of elements of M and ν ∈ N(I) then we let
γ ν (x) = × γ να (xα )
α∈I
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DAVID RYDH
which is an element of ΓdA (M ) with d = |ν| =
P
α∈I
να .
(1.1.3) Functoriality — ΓA (·) is a covariant functor from the category of Amodules to the category of graded A-algebras [Rob63, Ch. III §4, p. 251].
(1.1.4) Base change — If A0 is an A-algebra then there is a natural isomorphism ΓA (M ) ⊗A A0 → ΓA0 (M ⊗A A0 ) mapping γ d (x) ⊗A 1 to γ d (x ⊗A 1) [Rob63,
Thm. III.3, p. 262].
(1.1.5) Multiplicative structure — When B is an A-algebra then the multiplication of B induces a multiplication on ΓdA (B) which we will denote by juxtaposition [Rob80]. This multiplication is such that γ d (x)γ d (y) = γ d (xy).
(1.1.6) Universal property — If M is an A-module, then the A-module ΓdA (M )
represents polynomial laws which are homogeneous of degree d [Rob63, Thm. IV.1,
p. 266]. If B is an A-algebra, then the A-algebra ΓdA (B) represents multiplicative
polynomial laws which are homogeneous of degree d [Fer98, Prop. 2.5.1].
(1.1.7) Basis — If (xα )α∈I is a set of generators of M , then γ ν (x) ν∈N(I) is a set
of generators of ΓA (M ). If (xα )α∈I is a basis of M then γ ν (x) ν∈N(I) is a basis of
ΓA (M ) [Rob63, Thm. IV.2, p. 272].
(1.1.8) Presentation — Let M = G/R be a presentation of the A-module M .
Then ΓA (M ) = ΓA (G)/I where I is the ideal of ΓA (G) generated by the images in
ΓA (G) of γ d (x) for every x ∈ R and d ≥ 1 [Rob63, Prop. IV.8, p. 284].
(1.1.9) Γ and TS — The homogeneous polynomial law M → TSdA (M ) of degree d
given by x 7→ x⊗A d = x⊗A · · ·⊗A x corresponds by the universal property (1.1.6) to
an A-module homomorphism ϕ : ΓdA (M ) → TSdA (M ) [Rob63, Prop. III.1, p. 254].
When M is a free A-module then ϕ is an isomorphism [Rob63, Prop. IV.5, p. 272].
The functors ΓdA and TSdA commute with filtered direct limits [I, 1.1.4, 1.2.11]. Since
any flat A-module is the filtered direct limit of free A-modules [Laz69, Thm. 1.2], it
thus follows that ϕ : ΓdA (M ) → TSdA (M ) is an isomorphism for any flat A-module
M.
Moreover by [Rob63, Prop. III.3, p. 256], there is a diagram of A-modules
TSdA (M ) O
ΓdA (M )
o

/ Td (M )
A
SdA (M )
such that going around the square is multiplication by d!. Thus if d! is invertible
then ΓdA (M ) → TSdA (M ) is an isomorphism. In particular, this is the case when A
is purely of characteristic zero, i.e., contains the field of rationals.
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
5
Let B be an A-algebra. As the law B → TSdA (B) given by x 7→ x⊗A d is multiplicative, it follows that the homomorphism ϕ : ΓdA (B) → TSdA (B) is an A-algebra
homomorphism. In Section 4.1 we study the properties of ϕ more closely.
1.2. When is ΓdA (M ) generated by γ d (M )? ΓdA (M ) is not always generated by
γ d (M ) but a result due to Ferrand [Fer98, Lem. 2.3.1], cf. Proposition (1.2.4), shows
that there is a finite free base change A ,→ A0 such that ΓdA0 (M ⊗A A0 ) is generated
by γ d (M ⊗A A0 ). We
will prove a slightly stronger statement in Proposition (1.2.2).
We let γ d (M ) denote the A-submodule of ΓdA (M ) generated by the subset
γ d (M ).
Lemma (1.2.1). Let A be a ring and M an A-module. There is a commutative
diagram
d
/ Γd (M ) ⊗A A0
(M ) ⊗A A0
γA
A
ϕ
0
d
γA
0 (M ⊗A A )
◦
⊆
ψ ∼
=
ΓdA0 (M ⊗A A0 )
where ψ is the canonical isomorphism of (1.1.4). If A → A0 is a surjection or
a
0
d
=
localization then ϕ is surjective.
In
particular,
if
in
addition
γ
0 (M ⊗A A )
A
d
ΓdA0 (M ⊗A A0 ) then γA
(M ) ⊗A A0 → ΓdA (M ) ⊗A A0 is surjective.
Proof. The morphism ϕ is well-defined as ψ γ d (x) ⊗A a0 = a0 γ d (x ⊗A 1) if x ∈ M
and a0 ∈ A0 . If A0 = A/I then ϕ is clearly surjective. If A0 = S −1 A is a localization
0
then ϕ is surjective
any element
of M ⊗A A can be written as x ⊗A (1/f ) and
since
d
d
d
ϕ γ (x) ⊗A 1/f = γ x ⊗A (1/f ) .
Proposition (1.2.2). Let M be an A-module. The A-module ΓdA (M ) is generated
by the subset γ d (M ) if the following condition is satisfied
(*) For every p ∈ Spec(A) the residue field k(p) has at least d elements or Mp
is generated by one element.
If M is of finite type, then this condition is also necessary.
d
d
Proof. Lemma (1.2.1) gives that γA
(M ) = ΓdA (M ) if and only if γA
(Mp ) =
p
ΓdAp (Mp ) for every p ∈ Spec(A). We can thus assume that A is a local ring and
only need to consider the condition (*) for the maximal ideal m. If M is generated
d
by one element then it is obvious that γA
(M ) = ΓdA (M ).
d
Further, any element in ΓA (M ) is the image of an element in ΓdA (M 0 ) for some
submodule M 0 ⊆ M of finite type. It is thus sufficient, but not necessary, that
ΓdA (M 0 ) is generated by γ d (M 0 ) for every submodule M 0 ⊆ M of finite type. We can
thus assume that M is of finite type. Lemma (1.2.1) applied with A A/m = k(m)
d
together with Nakayama’s
lemma then shows that γA
(M ) = ΓdA (M ) if and only
d
if γA/m
(M/mM ) = ΓdA/m (M/mM ). We can thus assume that A = k is a field.
We will prove by induction on e that Γek (M ) is generated by γ e (M ) when 0 ≤
e ≤ d if and only if either rk M ≤ 1 or |k| ≥ e. Every element in Γek (M ) is a linear
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DAVID RYDH
combination of elements of the form
γ ν (x) = γ ν1 (x1 ) × γ ν2 (x2 ) × · · · × γ νm (xm ).
where xi ∈ M and |ν| = e. By induction γ ν2 (x2 )×· · ·×γ νm (xm ) ∈ γ e−ν1 (M ) and
we can thus assume that m = 2 and it is enough to show that γ i (x) × γ e−i (y) ∈
γ e (M ) for every x, y ∈ M and 0 ≤ i ≤ e if and only if either rk M ≤ 1 or
|k| ≥ e. If x and y are linearly dependent this is obvious. Thus we need to show
that for x and y linearly independent, we have that γ i (x) × γ e−i (y) ∈ γ e (kx ⊕ ky)
if and only if |k| ≥ e. A basis for Γek (kx ⊕ ky) is given by z0 , z1 , . . . , ze where
zi = γ i (x) × γ e−i (y), see (1.1.7). For any a, b ∈ k we let
ξa,b := γ e (ax + by) =
e
X
i=0
γ i (ax) × γ e−i (by) =
e
X
ai be−i zi .
i=0
P
Le
Then γke (kx ⊕ ky) = Γek (kx ⊕ ky) if and only if (a,b)∈k2 kξa,b = i=0 kzi . Since
P
L
e
ξλa,λb = λe ξa,b this is equivalent to (a:b)∈P1 kξa,b = i=0 kzi . It is thus necessary
k
1
that Pk = |k| + 1 ≥ e + 1. On the other hand if a1 , a2 , . . . , ae ∈ k ∗ are distinct
then ξa1 ,1 , ξa2 ,1 , . . . , ξae ,1 , ξ1,0 are linearly independent. In fact, this amounts to
(1, ai , a2i , a3i , . . . , aei )i=1,2,...,e and (0, 0, . . . , 0, 1) being linearly independent in k e+1 .
If they are dependent then there exist a non-zero (c0 , c1 , . . . , ce−1 ) ∈ k e such that
= 0 for every 1 ≤ i ≤ e but this is impossible
c0 + c1 ai + c2 a2i + · · · + ce−1 ae−1
i
since c0 + c1 x + · · · + ce−1 xe−1 = 0 has at most e − 1 solutions.
Lemma
(1.2.3). Let Λd = Z[T ]/Pd (T ) where Pd (T ) is the unitary polynomial
Q
i
j
0≤i<j≤d (T − T ) − 1. Then every residue field of Λd has at least d + 1 elements.
In particular, if A is any algebra, then A ,→ A0 = A ⊗Z Λd is a faithfully flat finite
extension such that every residue field of A0 has at least d + 1 elements.
Proof. The Vandermonde matrix (T ij )0≤i,j≤d is invertible in EndΛd (Λd+1
) since it
d
has determinant one. Let k be a field and ϕ : Λd → k be any homomorphism. If t =
ϕ(T ) then (tij )0≤i,j≤d is invertible in Endk (k d+1 ) and it follows that 1, t, t2 , . . . , td
are all distinct and hence that k has at least d + 1 elements.
Proposition (1.2.4). [Fer98, Lem. 2.3.1] Let Λd be as in Lemma (1.2.3). If A is
a Λd -algebra then ΓdA (M ) is generated by γ d (M ). In particular, for every A there
is a finite faithfully flat extension A → A0 , independent of M , such that ΓdA0 (M 0 )
is generated by γ d (M 0 ).
Proof. Follows immediately from Proposition (1.2.2) and Lemma (1.2.3).
1.3. Generators of the ring of divided powers. In this section we will recall
some results on the degree of the generators of ΓdA (B). For our purposes the results
of Fleischmann [Fle98] is sufficient and we will not use the more precise and stronger
statements of [Ryd07a] even though some bounds would be slightly improved then.
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
7
Definition (1.3.1) (Multidegree). Let B = A[x1 , x2 , . . . , xr ]. We define the multidegree of a monomial xα ∈ B to be α. This makes B into a Nr -graded ring
M
M
B=
Bα =
Axα
α∈Nr
α∈Nr
Let M be the A-module basis of B consisting of the monomials. Recall from
paragraph (1.1.7) that a basis of ΓA (B)
is given by the elements γ ν (x) = ×α γ να (xα )
for ν ∈ N(M) . We let mdeg γ k (xα ) = kα and mdeg(f × g) = mdeg(f ) + mdeg(g)
for f, g ∈ ΓA (B). Then
X
X
mdeg × γ να (xα ) =
να mdeg(xα ) =
να α.
α
xα ∈M
α∈Nr
We let ΓdA (B)α be the A-module
generated by basis elements γ ν (x) of multidegree
L
d
α. This makes ΓA (B) = α∈Nr ΓdA (B)α into a Nr -graded ring.
L
Definition (1.3.2) (Degree). Let B = A[x1 , x2 , . . . , xr ] = k≥0 Bk with the usual
grading, i.e., L
Bk are the homogeneous polynomials of degree k. The graded Ad
d
d
algebra C =
k≥0 ΓA (Bk ) is a subalgebra of ΓA (B). If an element f ∈ ΓA (B)
belongs to Ck = ΓdA (Bk ) we say that f is homogeneous of degree k. The degree
of an arbitrary
element f ∈ ΓdA (B) is the smallest natural number n such that
Ln
d
f ∈ ΓA ( k=0 Bk ).
L
d
Remark (1.3.3). Let B = A[x0 , x1 , . . . , xr ] and let C =
k≥0 ΓA (Bk ) be the
d
graded subring of ΓA (B). The degree in the previous definition is such that there
is a relation between the degree of elements in C and the degree of an element in
the graded localization C(γ d (s)) for s ∈ B1 . To see this, note that
C(γ d (s)) = ΓdA B(s) = ΓdA A[x0 /s, . . . , xr /s] .
We let A[x0 /s, . . . , xr /s] be graded such that xi /s has degree 1. An element
f ∈ ΓdA (A[x0 /s, . . . , xr /s]) of degree n can then be written as g/γ d (s)n where
g ∈ ΓdA (Bn ) is homogeneous of degree n.
Theorem (1.3.4) ([Ric96, Prop. 2], [Ryd07a, Cor. 8.4]). If d! is invertible in A
then ΓdA (A[x1 , . . . , xr ]) is generated by the elementary multisymmetric functions
γ d1 (x1 )×γ d2 (x2 )×· · ·×γ dr (xr )×γ d−d1 −d2 −···−dr (1), di ∈ N and d1 +d2 +· · ·+dr ≤ d.
Theorem (1.3.5) ([Fle98, Thm. 4.6, 4.7], [Ryd07a, Cor. 8.6]). For an arbitrary
ring A, the A-algebra ΓdA (A[x1 , . . . , xr ]) is generated by γ d (x1 ), γ d (x2 ), . . . , γ d (xr )
and the elements γ k (xα )×γ d−k (1) with kα ≤ (d−1, d−1, . . . , d−1). Further, there
is no smaller multidegree bound and if d = ps for some prime p not invertible in A,
then ΓdA (A[x1 , . . . , xr ]) is not generated by elements of strictly smaller multidegree.
Theorems (1.3.4) and (1.3.5) give the following degree bound:
8
DAVID RYDH
Corollary (1.3.6). Let A be a ring and B = k[x1 , x2, . . . , xr ]. Then ΓdA (B) is
generated by elements of degree at most max 1, r(d − 1) . If d! is invertible in A,
then ΓdA (B) is generated by elements of degree one.
2. Weighted projective schemes and quotients by finite groups
In this section we review the definition and the basic results on weighted projective schemes. We will in particular focus on weighted projective structures which
are covered in degree one. By this, we mean that the sections of O(1) give an affine
cover of the weighted projective scheme. We then recall the construction, using
invariant theory, of a geometric quotient of a projective scheme by a finite group.
Finally, we discuss the failure of a geometric quotient to commute with arbitrary
base change and closed immersions.
2.1. Remarks on projectivity. We will follow the definitions in EGA. In particular, very ample, ample, quasi-projective and projective will have the meanings
of [EGAII , §4.4, §4.6, §5.3, §5.5]. By definition, a morphism q : X → S is quasiprojective if it is of finite type and there exists an invertible OX -sheaf L ample
with respect to q. Note that this does not imply that X is a subscheme of PS (E)
for some quasi-coherent OS -module E. However, if S is quasi-compact and quasiseparated then there is a quasi-coherent OS -module of finite type E and an immersion X ,→ P(E) [EGAII , Prop. 5.3.2]. Similarly, a projective morphism is always
quasi-projective and proper but the converse only holds if S is quasi-compact and
quasi-separated.
Furthermore, if q : X → S is a projective morphism and L a very ample
invertible sheaf on X then L does not necessarily correspond to a closed embedding
into a projective space over S. We always have a closed embedding X ,→ P(q∗ L)
as q is proper [EGAII , Prop. 4.4.4] but q∗ L need not be of finite type. If S is
locally noetherian however, then q∗ L is of finite type [EGAIII , Thm. 3.2.1]. If S
is quasi-compact and quasi-separated then we can find a sub-OS -module of finite
type E of q∗ L such that we have a closed immersion i : X ,→ P(E) and such that
L = i∗ OP(E) (1).
We will also need the following stronger notion of projectivity introduced by
Altman and Kleiman in [AK80, §2]. Our definition differs slightly from theirs as
we do not require strongly projective morphisms to be of finite presentation.
Definition (2.1.1). A morphism X → S is strongly projective (resp. strongly
quasi-projective) if it is of finite type and factors through a closed immersion (resp.
an immersion) X ,→ PS (L) where L is a locally free OS -module of constant rank.
Remark (2.1.2). A strongly (quasi-)projective morphism is (quasi-)projective and
the converse holds when S is quasi-compact, quasi-separated and admits an ample
sheaf, e.g., S affine [AK80, Ex. 2.2 (i)]. In fact, in this case there is an embedding
X ,→ PnS and thus the notions of projective and strongly projective also agree with
the definitions in [Har77].
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
9
2.2. Weighted projective schemes.
Definition (2.2.1). Let S be a scheme. A weighted projective scheme over S is an
S-scheme X together with a quasi-coherent graded OS -algebra A of finite type, not
necessarily generated by degree one elements, such that X = ProjS (A). We let as
] for any n ∈ Z.
usual OX (n) = A(n)
If A is generated by degree one elements then the sheaves OX (n) are invertible
for any integer n and very ample if n is positive. Furthermore, there is then a
canonical isomorphism OX (m) ⊗OX OX (n) = OX (m + n). All these properties
may fail if A is not generated by degree one elements.
It can however be shown, cf. Corollary (2.2.4), that if S is quasi-compact then
q : X → S is projective. To be precise, there is a positive integer n such that OX (n)
is invertible, the homomorphism q ∗ An → OX (n) is surjective and in : X → P(An )
is a closed immersion. In particular, OX (n) = i∗n OP(An ) (1) is very ample. Another
consequence is that if X is a weighted projective scheme over an arbitrary scheme
S then X → S is proper.
We will give a demonstration of the projectivity of X → S when S is quasicompact and also show some properties of the sheaves OX (n). The results are
somewhat weaker than those in [BR86, §4] but we also give stronger results when
X is covered in degree one.
The following lemma is an explicit form of [EGAII , Lem. 2.1.6].
Lemma (2.2.2). If B is a graded A-algebra generated by elements f1 , f2 , . . . , fs ∈
B of degrees d1 , d2 , . . . , ds and l is the least common multiple of d1 , d2 , . . . , ds then
(i) Bn+l = (Bn Bl ) for every n ≥ (s − 1)(l − 1).
(ii) Bkn = (Bn )k for every k ≥ 0 if n = al with a ≥ s − 1.
P
Proof. Clearly Bk is generated by f1a1 f2a2 . . . fsas such that i ai di = k. Let gi =
l/d
fi i ∈ Bl . If k ≥ s(l − 1) + 1 and f = f1a1 f2a2 . . . fsas ∈ Bk then gi |f for some i
which shows (i). (ii) follows easily from (i).
Proposition (2.2.3) (cf. [BR86, Cor. 4A.5, Thm. 4B.7]). Let A be a ring and
let B be a graded A-algebra generated by a finite number of elements f1 , f2 , . . . , fs
of degrees d1 , d2 , . . . , ds . Let l be the least common multiple of the di ’s. Let S =
] Then
Spec(A), X = Proj(B) and OX (n) = B(n).
S
(i) X = f ∈Bn D+ (f ) if n = al and a ≥ 1.
(ii) OX (n) is invertible if n = al and a ∈ Z.
(iii) OX (n) is ample and generated by global sections if n = al and a ≥ 1.
(iv) The canonical homomorphism OX (al) ⊗ OX (n) → OX (al + n) is an isomorphism for every a, n ∈ Z.
(v) If n = al with a ≥ 1 then there is a canonical morphism in : X →
P(Bn ). If a ≥ max {1, s − 1} then in is a closed immersion and OX (n) =
i∗n OP(Bn ) (1) is very ample relative to S.
(vi) OX (n) is generated by global sections if n ≥ (s − 1)(l − 1).
10
DAVID RYDH
Ss
Ss
al/d Proof. (i) is trivial as X = i=1 D+ (fi ) = i=1 D+ fi i if a ≥ 1, cf [EGAII ,
Cor. 2.3.14]. Note that if f ∈ Bl then
(2.2.3.1)
Bf = B(f ) ⊕ B(1)(f ) ⊕ · · · ⊕ B(l − 1)(f ) [f, f −1 ].
Thus Γ D+ (f ), OX (al) = B(al)(f ) = B(f ) f a is a free B(f ) -module of rank one
which shows (ii).
(iii) If a ≥ 1 then D+ (f ) f ∈B is an affine cover of X. As OX (al) is an invertible
al
sheaf it is thus generated by global sections and ample by definition, cf. [EGAII ,
Def. 4.5.3 and Thm. 4.5.2 a0 )].
(iv) It is enough to show that the homomorphism OX (al)⊗OX (n) → OX (al +n)
is an isomorphism locally over D+ (f ) with f ∈ Bl . Locally this homomorphism is
B(al)(f ) ⊗B(f ) B(n)(f ) → B(al+n)(f ) which is an isomorphism by equation (2.2.3.1)
(v) If n = al with a ≥ 1 then by (i) the morphism in : X → P(Bn ) is everywhere
defined. If in addition a ≥ s − 1 then B (n) is generated by degree one elements by
Lemma (2.2.2) (ii). Thus we have a closed immersion X = Proj(B) ∼
= Proj(B (n) ) ,→
P(Bn ).
(vi) Assume that n ≥ (s − 1)(l − 1), then Bn+kl = (Bn Blk ) for any positive
0
k
integer k by Lemma (2.2.2) (i). If f ∈ Bl and
b ∈ B(n)(f ) , then b = b /f for some
k
0
b ∈ Bn+kl = Bn Bl and thus b ∈ B(f ) Bn . This shows that OX (n) is generated
by global sections as Bn ⊆ Γ D+ (f ), OX (n)).
Corollary (2.2.4) ([EGAII , Cor. 3.1.11]). If S is quasi-compact and X = ProjS (A)
is a weighted projective scheme then there exists a positive integer n such that
X → P(An ) is everywhere defined and a closed immersion. In particular X is
projective and OX (n) is very ample relative to S.
Proof. Let {Si } be a finite affine cover of S and let Ai = Γ(Si , OS ) and Bi =
Γ(Si , A). Then as Bi is a finitely generated graded Ai -algebra, there
is by Propo
sition (2.2.3) a positive integer ni such that X ×S Si → P (Bi )ni is defined and a
closed immersion. Choosing n as the least common multiple of the ni ’s we obtain
a closed immersion X ,→ P(An ).
Remark (2.2.5). Note that (2.2.3) (iv), (v), (vi) implies that the following are
equivalent:
(i) OX (n) is invertible for all 0 < n < l.
(ii) OX (n) is invertible for all n.
(iii) OX (n) is very ample for all sufficiently large n.
As (i) is easily seen to not hold in many examples in particular (iii) is not always
true.
The following condition will be important later on as it is satisfied for Symd (X/S)
for X/S quasi-projective. Note that in the remainder of this section we do not
assume that A is finitely generated. In particular, ProjS (A) need not be a weighted
projective space.
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
11
Definition (2.2.6). Let S be a scheme, A a graded quasi-coherent OS -algebra and
X=
S ProjS (A). If there is an affine cover (Sα ) of S such that X ×S Sα is covered
by f ∈Γ(Sα ,A1 ) D+ (f ), then we say that X/S is covered in degree one.
Proposition (2.2.7). Let A be a ring and let B be a graded A-algebra generated
] If
by elements of degree ≤ d. Let S = Spec(A), X = Proj(B) and OX (n) = B(n).
X/S is covered in degree one then
S
(i) X = f ∈Bn D+ (f ) if n ≥ 1.
(ii) OX (n) is invertible for n ∈ Z and ample and generated by global sections
if n ≥ 1.
(iii) OX (m) ⊗ OX (n) ∼
= OX (m + n) for every m, n ∈ Z.
(iv) The canonical morphism in : X → P(Bn ) is defined for every n ≥ 1. If
n ≥ d then in is a closed immersion and OX (n) = i∗n OP(Bn ) (1) is very
ample relative S.
Proof. (i) is equivalent to X/S being covered
in degree one. Using the cover X =
S
S
f ∈B1 D+ (f ) instead of the cover X =
f ∈Bl D+ (f ) we may then prove (ii) and (iii)
exactly as (ii), (iii) and (iv) in Proposition (2.2.3).
(iv) Let n ≥ d and let B 0 be the sub-A-algebra of B generated by Bn . It is enough
to show that the inclusion B 0 ,→ B induces
Proj(B) ∼
= Proj(B 0 ).
S an isomorphism
n
We will show this using the cover X = f ∈B1 D+ (f ). Let f ∈ B1 and g ∈ B(f n )
0
such that g = b/f nk for some b ∈ Bnk . To show that g ∈ B(f
n ) we can assume that
b = b1 b2 . . . bs is a product ofQ
elements of degrees
d
≤
d,
as
every element of Bnk
ns i
s
n−di
0
are sums of such. Then g =
b
f
/f
which
is
an
element of B(f
n).
i=1 i
Corollary (2.2.8). Let S be any scheme and A a graded quasi-coherent OS -algebra
such that A is generated by elements of degree at most d. Let X = Proj(A),
] and assume that X/S is covered in degree one. Then
OX (n) = A(n)
(i) OX (n) = OX (1)⊗n is invertible for every n ∈ Z.
(ii) If n ≥ 1 then OX (n) is ample and q ∗ An → OX (n) is surjective.
(iii) For every n ≥ 1 the canonical morphism in : X → P(An ) is everywhere
defined. If n ≥ d it is a closed immersion.
In particular, if X = ProjS (A) also is a weighted projective scheme, i.e., if A is of
finite type, then X is projective.
Example (2.2.9) (Standard weighted projective spaces). Let A = k be an algebraically closed field of characteristic zero and B = k[x0 , x1 , . . . , xr ]. Choose
positive integers d0 , d1 , . . . , dr and consider the action of G = µd0 × · · · × µdr ∼
=
Z/d0 Z × · · · × Z/dr Z on B given by (n0 , n1 , . . . , nr ) · xi = ξdnii xi where ξdi is a di th
primitive root of unity. Then B G = k[xd00 , xd11 , . . . , xdr r ] and Proj B G is a weighted
projective space of type (d0 , d1 , . . . , dr ).
It can be seen, cf. Proposition (2.3.4), that Proj B G is the geometric quotient
of Proj(B) = Pr by G. More generally, if S is a noetherian scheme and X/S is
12
DAVID RYDH
projective with an action of a finite group G linear with respect to a very ample
sheaf OX (1), then a geometric quotient X/G exists and can be given a structure
as a weighted projective scheme.
The weighted projective space Proj B G is often denoted P(d0 , d1 , . . . , dr ). It
can also be constructed as the quotient of Ar+1 − 0 by Gm where Gm acts on Ar+1
by λ · xi = λdi xi . The closed points of P(d0 , d1 , . . . , dr ) are thus {x = (x0 : x1 : · · · :
xr )} = k r+1 / ∼ where x ∼ y if there is a λ ∈ k ∗ such that λdi xi = yi for every i.
2.3. Quotients of projective schemes by finite groups. Let X be an S-scheme
and G a discrete group acting on X/S, i.e., there is a group homomorphism G →
AutS (X). In the category of ringed spaces we can construct a quotient Y = (X/G)rs
as following. Let Y as a topological space be X/G with the quotient topology, and
quotient map q : X → Y . Further let the sheaf of sections OY be the subsheaf
G
(q∗ OX ) ,→ q∗ OX of G-invariant sections. Note that G acts on q∗ OX since for any
open subset U ⊆ Y the inverse image q −1 (U ) is G-stable and hence has an induced
action of G. Thus we obtain a ringed S-space (Y, OY ) together with a morphism
of ringed S-spaces q : X → Y . The ringed space (Y, OY ) is not always a scheme,
in fact not always even a locally ringed space. But when it exists as a scheme it is
called the geometrical quotient and is also the categorial quotient in the category of
schemes over S. For general existence results we refer to [Ryd07b]. The existence of
a geometric quotient of an affine schemes by a finite group is not difficult to show:
Proposition (2.3.1) ([SGA1 , Exp. V, Prop. 1.1, Cor. 1.5]). Let S be a scheme, A
a quasi-coherent sheaf of OS -algebras and X = SpecS (A). An action of G on
X/S
induces an action of G on A. If G is a finite group then Y = SpecS AG is the
geometric quotient of X by G. If S is locally noetherian and X → S is of finite
type, then Y → S is of finite type.
From this local result it is not difficult to show the following result:
Theorem (2.3.2) ([SGA1 , Exp. V, Prop. 1.8]). Let f : X → S be a morphism
of arbitrary schemes and G a finite discrete group acting on X by S-morphisms.
Assume that every G-orbit of X is contained in an affine open subset. Then the
geometrical quotient q : X → Y = X/G exists as a scheme.
It can also be shown, from general existence results, that if X/S is separated
then this is also a necessary condition [Ryd07b, Rmk. 4.9].
Remark (2.3.3). If X → S is quasi-projective, then every G-orbit is contained in
an affine open set. In fact, we can assume that S = Spec(A) is affine and thus
that we have an embedding X ,→ PnS . For any orbit Gx we can then choose a
section f ∈ OPn (m) for some sufficiently large m such that V(f ) does not intersect
Gx. The affine subset D(f ) then contains the orbit Gx. More generally [EGAII ,
Cor. 4.5.4] shows that every finite set, in particulary every G-orbit, is contained in
an affine open set if X/S is such that there exists an ample invertible sheaf on X
relative to S.
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
13
In Corollary (2.3.6) we will show that if S is a noetherian scheme and X → S
is (quasi-)projective, then so is X/G → S. In fact if X is projective we will give a
weighted projective structure on X/G.
L
Proposition (2.3.4). Let S be a scheme and let A = d≥0 Ad be a graded quasicoherent OS -algebra, generated by degree one elements. Let G be a finite group
acting on A by graded OS -algebra automorphisms. Then G acts on X = ProjS (A)
linearly with respect to OX (1). As X admits a very ample invertible sheaf relative
to S, a geometric quotient Y = X/G exists, cf. Remark (2.3.3). There is an
isomorphism Y ∼
= ProjS AG and under this isomorphism, the quotient map q :
X → Y is induced by AG ,→ A.
e
Proof. Everything is local over S so we can assume that S = Spec(A), A = B
and X = Proj(B). We can cover X by G-stable affine subsets of the form D+ (f )
with f ∈ B G homogeneous. In fact, if Z is a G-orbit of X then the demonstration
of [EGAII , Cor. 4.5.4] Q
shows that there is a homogeneous f 0 ∈ B such that Z ⊆
D+ (f 0 ). If we let f = σ∈G σ(f 0 ), then Z ⊆ D+ (f ) and f ∈ B G is homogeneous.
Over such an open set we have that
G X|D+ (f ) /G = Spec B(f )
= Spec (B G )(f ) = Proj B G |D+ (f ) .
It is thus clear that Y = Proj B G .
Remark (2.3.5). Note that AG is not always generated by AG
1 even though A is
e we may not be able to
generated by A1 . Also, if S = Spec(A) is affine and A = B,
cover X = Proj(B) with G-stable affine subsets of the form D+ (f ) with f ∈ B1G .
This is demonstrated by example (2.2.9) if we choose di > 1 for some i.
Corollary (2.3.6) ([Knu71, Ch. IV, Prop. 1.5]). Let S be noetherian, X → S
be projective (resp. quasi-projective) and G a finite group acting on X by Smorphisms. Then the geometrical quotient X/G is projective (resp. quasi-projective).
Proof. Let X ,→ (X/S)m = X ×S X ×S · · · ×S X be the closed immersion given
by x → (σ1 x, σ2 x, . . . , σm x) where G = {σ1 , σ2 , . . . , σm }. As X → S is quasiprojective and S is noetherian, there is an immersion X ,→ PS (E) for some quasicoherent OS -module of finite type E, see [EGAII , Prop. 5.3.2]. This immersion
m
together with the immersion
n X ,→ (X/S) given above, gives a G-equivariant
n
immersion X ,→ PS (E)/S if we let G permute the factors of PS (E)/S . Following this immersion by the Segre embedding we get a G-equivariant immersion
f : X ,→ PS (E ⊗m ) where G acts linearly on PS (E ⊗m ), i.e., by automorphisms of
E ⊗m .
Let Y = f (X) be the schematic image of f . As Y is clearly G-stable we have an
action of G on Y and a geometric quotient q : Y → Y /G. Then, as X ,→ Y is an
open immersion and q is open, we have that X/G = (Y /G)|q(Y ) . Thus it is enough
to show that Y /G is projective. Let A = S(E ⊗m )/I such that Y = Proj(A). Then
there is an action of G on A1 which induces the action Y . By Proposition (2.3.4)
14
DAVID RYDH
we have that Y /G = Proj AG . The scheme Y /G is a weighted projective scheme
as AG is an OS -algebra of finite type by Proposition (2.3.1). It then follows by
Corollary (2.2.4) that Y /G is projective.
2.4. Finite quotients, base change and closed subschemes. A geometric quotient is always uniform, i.e., it commutes with flat base change [GIT, Rmk. (7), p. 9].
It is also a universal topological quotient, i.e., the fibers corresponds to the orbits
and the quotient has the quotient topology and this holds after any base change.
However, in positive characteristic a geometric quotient is not necessarily a universal geometric quotient, i.e., it need not commute with arbitrary base change. This
is shown by the following example:
Example (2.4.1). Let X = Spec(B), S = Spec(A), S 0 = Spec(A/I) with A =
k[]/2 where k is a field of characteristic p > 0, B = k[, x]/(2 , x) and I = ().
We have an action of G = Z/p = hτ i on B given by τ (x) = x + and τ () = .
Then τ (xn ) = xn for all n ≥ 2 and thus B G = k[, x2 , x3 ]/(2 , x2 , x3 ). Further,
we have that (B ⊗A A0 )G = k[x] and B G ⊗A A0 = k[x2 , x3 ].
Recall that a morphism of schemes is a universal homeomorphism if the underlying morphism of topological spaces is a homeomorphism after any base change.
Proposition (2.4.2) ([EGAIV , Cor. 18.12.11]). Let f : X → Y be a morphism
of schemes. Then f is a universal homeomorphism if and only if f is integral,
universally injective and surjective.
Proposition (2.4.3). Let X/S be a scheme with an action of a finite group G such
that every G-orbit of X is contained in an affine open subset. Let S 0 → S be any
morphism and let X 0 = X ×S S 0 . Then geometric quotients q : X → X/G and
r : X 0 → X 0 /G exists. Let (X/G)0 = (X/G) ×S S 0 . As r is a categorical quotient
we have a canonical morphism X 0 /G → (X/G)0 . This morphism is a universal
homeomorphism.
Proof. The geometric quotients q and r exists by Theorem (2.3.2). As q and r
are universal topological quotients it follows that X 0 /G → (X/G)0 is universally
bijective. As X 0 → X 0 /G is surjective and X 0 → (X/G)0 is universally open it
follows that X 0 /G → (X/G)0 is universally open and hence a universal homeomorphism.
If G acts on X and U ⊆ X is a G-stable open subscheme, then U/G is an
open subscheme of X/G. In fact, U/G is the image of U by the open morphism
q : X → X/G. If Z ,→ X is a closed G-stable subscheme, then Z/G is not always
the image of Z by q. In fact, Z/G need not even be a subscheme of X/G. We have
the following result:
Proposition (2.4.4). Let G be a finite group, X/S a scheme with an action of
G such that the geometric quotient q : X → X/G exists. Let Z ,→ X be a
closed G-stable subscheme. Then the geometric quotient r : Z → Z/G exist. Let
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
15
q(Z) be the scheme-theoretic image of the morphism Z ,→ X → X/G. As r is
a categorical quotient, the morphism Z → q(Z) ,→ X/G factors canonically as
Z → Z/G → q(Z) ,→ X/G. The morphism Z/G → q(Z) is a schematically
dominant universal homeomorphism.
Proof. As Z/G and q(Z) both are universal topological quotients of Z, the canonical
morphism Z/G → q(Z) is universally bijective. Since Z → q(Z) is universally open
and Z → Z/G is surjective we have that Z/G → q(Z) is universally open and thus
a universal homeomorphism. Further as Z → q(Z) is schematically dominant the
morphism Z/G → q(Z) is also schematically dominant.
Corollary (2.4.5). Let G and X/S be as in Proposition (2.4.4). There is a canonical universal homeomorphism (Xred )/G → (X/G)red .
We can say even more about the exact structure of Z/G → q(Z). For ease of
presentation we state the result in the affine case.
Proposition (2.4.6). Let A be a ring with an action by a finite group G and
let I ⊂ A be a G-stable
ideal. Let X = Spec(A)
and Z = Spec(A/I). Then
Z/G = Spec (A/I)G and q(Z) = Spec AG /I G . We have an injection AG /I G ,→
(A/I)G . If f ∈ (A/I)G then there is an n | card(G) such that f n ∈ AG /I G . To be
more precise we have that
(i) If A is a Z(p) -algebra with p a prime, e.g., a local ring with residue field k
or a k-algebra with char k = p, then n can be chosen as a power of p.
(ii) If A is purely of characteristic zero, i.e., a Q-algebra, then AG /I G ,→
(A/I)G is an isomorphism.
Proof. Let f ∈ A such that its image f ∈ A/I is G-invariant. To show that
n
n
f ∈ AG /I G for some positive integer n it is enough to show that f ∈ AG /I G ⊗Z Zp
for every p ∈ Spec(Z). As Z → Zp is flat, we have that
AG ⊗Z Zp = (A ⊗Z Zp )G
I G ⊗Z Zp = (I ⊗Z Zp )G
AG /I G ⊗Z Zp = (A ⊗Z Zp )G /(I ⊗Z Zp )G
(A/I)G ⊗Z Zp = (A/I ⊗Z Zp )G .
Thus we can assume that A is a Zp -algebra.
Let q be the characteristic exponent of Zp /pZp , i.e., q = p if p = (p), p > 0 and
q = 1 if p = (0). Choose positive integers k and m such that card(G) = q k m and
q - m if q 6= 1. Then choose a Sylow subgroup H of G of order q k , or H = (e) if
q = 1, and let σ1 H, σ2 H, . . . , σm H be its cosets. Then
g=
m
1 X Y
σ(f )
m i=1
σ∈σi H
is G-invariant and its image g ∈ AG /I G maps to f
qk
∈ (A/I)G .
16
DAVID RYDH
Proposition (2.4.6) can also be extended to the case where G is any reductive
group [GIT, Lem. A.1.2].
Remark (2.4.7). Let X/S be a scheme with an action of a finite group G with
geometric quotient q : X → X/G. Then
(i) If S is a Q-scheme and S 0 → S is any morphism then (X ×S S 0 )/G =
X/G ×S S 0 .
(ii) If S is arbitrary and U ⊆ X is an open immersion then U/G = q(U ).
(iii) If S is a Q-scheme and Z ,→ X is a closed immersion then Z/G = q(Z).
(iv) If S is a Q-scheme then (X/G)red = Xred /G.
(ii) follows from the uniformity of geometric quotients, (i) and (iv) follows from
the universality of geometric quotients in characteristic zero and (iii) follows from
Proposition (2.4.6).
Statement (iii) can also be proven as follows. We can assume that X = Spec(A)
is affine. Then the homomorphism AG ,→ APhas an AG -module retraction, the
1
G
Reynolds-operator R, given by R(a) = card(G)
σ∈G σ(a). This implies that A ,→
A is universally injective, i.e., injective after tensoring with any A-module M . In
G
G
G
particular AG ,→ A is cyclically pure,
I = I ∩ A , for any
i.e., I0 A = I, where
G
G
G
then Z = X ×S S 0 =
ideal I ⊆ A. If we let S = Spec A and S = Spec A /I
Spec(A/I) and (iii) follows from (i).
3. The parameter spaces
In this section we define the symmetric product Symd (X/S), the scheme of divided powers Γd (X/S), and the Chow scheme of zero-cycles Chow0,d (X ,→ P(E)).
We show that when X/S is a projective bundle, then Symd (X/S) is projective and
we essentially obtain a bound on the generators of the natural weighted projective
structure, cf. Theorem (3.1.12). As a corollary, we show that if X/S is projective then so is Γd (X/S), cf. Theorem
L (3.2.10). If A is a graded OS -algebra and
X = Proj(A) then Γd (X/S) = Proj( k≥0 Γd (Ak )) in analogy with the description
L
Symd (X) = Proj( k≥0 TSd (Ak )). The Chow scheme of X = Proj(A) is defined as
L
the projective scheme corresponding to the subalgebra of k≥0 Γd (Ak ) generated
by degree one elements. The reduction of this scheme is the classical Chow variety.
3.1. The symmetric product.
Definition (3.1.1). Let X be a scheme over S and d a positive integer. We let
the symmetric group on d letters Sd act by permutations on (X/S)d = X ×S X ×S
· · · ×S X. When X/S is separated, the geometric quotient of (X/S)d by the action
of Sd exists as an algebraic space [Ryd07b] and we denote it by Symd (X/S) :=
(X/S)d /Sd . The scheme (or algebraic space) Symd (X/S) is called the dth symmetric product of X over S and is also denoted Symmd (X/S), (X/S)(d) or X (d) by
some authors.
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
17
Definition (3.1.2). Let X/S be a scheme. We say that X/S is AF if the following
condition is satisfied:
(AF)
Every finite set of points x1 , x2 , . . . , xn over the same point s ∈ S
is contained in an affine open subset of X.
Remark (3.1.3). If X has an ample sheaf relative to S, then X/S is AF, cf. [EGAII ,
Cor. 4.5.4]. It is also clear from [EGAII , Cor. 4.5.4] that if X/S is AF then so is
X ×S S 0 /S 0 for any base change S 0 → S. It can further be seen that if X/S is AF
then X/S is separated.
Remark (3.1.4). Let X/S be an AF-scheme and let d be a positive integer. By
Theorem (2.3.2) the geometric quotient Symd (X/S) is then a scheme. Let (Sα )
be an affine cover of S and let (Uαβ ) be an affine cover of X ×S Sα such that
any set of d points of X lying over the same point s ∈ Sα is included in some
U . Then (Uαβ /Sα )d is an open cover of (X/S)d by affine schemes. Thus
`αβ
d
d
α,β Sym (Uαβ /Sα ) → Sym (X/S) is an open covering by affines.
In the remainder of this section we will study the symmetric product when
S = Spec(A) is an affine scheme and X/S is projective. We will use the following
notation:
L
Notation (3.1.5). Let A be a ring and let B = k≥0 Bk be a graded A-algebra
finitely generated by elements in degree one. Let S = Spec(A) and X = Proj(B)
]
with very ample sheaf OL
X (1) = B(1) and canonical morphism q : X → S.
d
d
d
T
Further we let C =
k≥0 A (Bk ) ⊂ TA (B). Then (X/S) = Proj(C) and
d
Proj(C) ,→ P(C1 ) = P TA (B1 ) is the Segre embedding of (X/S)d corresponding
to the embedding X = Proj(B) ,→ P(B1 ). The permutation of the factors induces
L
an action of the symmetric group Sd on C and we let D = C Sd = k≥0 TSdA (Bk )
be the graded invariant ring.
By Proposition (2.3.4) we have that Symd (X/S) := Proj(C)/Sd = Proj(D). If A
is noetherian, then D is finitely generated and Symd (X/S) is a weighted projective
space. In general, however, we do not know that D is finitely generated. We
do know that Symd (X/S) → S is universally closed though, as (X/S)d → S is
projective.
Lemma (3.1.6). Let x1 , x2 , . . . , xd ∈ X be points such that q(x1 ) = q(x2 ) = · · · =
q(xd ) = s. Then there exists a positive integer n and an element f ∈ Bn ⊆
Γ X, OX (n) such that x1 , x2 , . . . , xd ∈ Xf = D+ (f ). If the residue field k(s) has
at least d elements then it is possible to take n = 1.
Proof. The existence of f for some n follows from [EGAII , Cor. 4.5.4]. For the last
assertion, assume that k(s) has at least d elements. As we can lift any element
f ∈ Bn ⊗A k(s) to an element f ∈ Bn after multiplying with an invertible element
of k(s), we can assume that A = k(s). Replacing B with the symmetric algebra
18
DAVID RYDH
S(B1 ) = k[t0 , t1 , . . . , tr ] we can further assume that B is a polynomial ring and
X = Prk(s) .
An element of B1 = Γ X, OX (1) is then a linear form f = a0 t0 +a1 t1 +· · ·+ar tr
with ai ∈ k(s) and can be thought of as a k(s)-rational point of (Prk(s) )∨ . The linear
forms which are zero in one of the xi ’s form a proper closed linear subset of all linear
forms. Thus if k(s) is infinite then there is a k(s)-rational point corresponding to
a linear form non-zero in every xi . If k = k(s) is finite, then at most (|k|r − 1)/|k ∗ |
linear forms are zero at a certain xi and equality holds when xi is k-rational. Thus
at most
d(|k|r − 1)/(|k| − 1) ≤ (|k|r+1 − |k|)/(|k| − 1)
= (|k|r+1 − 1)/(|k| − 1) − 1
linear forms contain at least one of the x1 , x2 , . . . , xd and hence there is at least one
linear form which does not vanish on any of the points.
Proposition (3.1.7). The product X d = X×S X×S · · ·×S X is covered by Sd -stable
affine open subsets of the form Xf ×S Xf ×S · · ·×S Xf where f ∈ Bn ⊆ Γ X, OX (n)
for some n. If every residue field of S has at least d elements then the open subsets
with f ∈ B1 ⊆ Γ X, OX (1) cover X d .
Proof. Follows immediately from Lemma (3.1.6).
Corollary (3.1.8).The symmetric product Symd (X/S) is covered by open affine
subsets Symd Xf /S with f ∈ Bn ⊆ Γ X, OX (n) for some n. If every residue field
of S has at least d elements then those affine subsets with n = 1 cover Symd (X/S).
d
Corollary (3.1.9). The symmetric
product Y = Sym (X/S) = Proj(D) is covered
by Yg where g ∈ D1 ⊆ Γ Y, OY (1) , i.e., Y = Proj(D) is covered in degree one.
Proof. Let A ,→ A0 be a finite flat extension such that every residue field of A0 has
at least d elements, e.g., the extension A0 = A ⊗Z Λd suffices by Lemma (1.2.3). Let
L
B 0 = B ⊗A A0 and C 0 = C ⊗A A0 and let D0 = D ⊗A A0 = n≥0 TSdA (Bn ) ⊗A A0 .
L
d
0
0
0
0
0
Then D0 =
n≥0 TSA0 (Bn ) as A ,→ A is flat. Note that if f ∈ Bn then g =
d
d
0
0
0
0
0
0
0
f ⊗f ⊗· · ·⊗f ∈ Dn and Sym (Xf 0 /S) = D+ (g ) as open subsets of Sym (X /S 0 ).
p
0 = D 0 . As Spec(A0 ) → Spec(A) is
Thus Corollary (3.1.8) shows that D10 D+
+
p
surjective it follows that D1 D+ = D+ .
We now use the degree bound on the generators of TSdA (A[x1 , . . . , xr ]) obtained
in Corollary (1.3.6) to get something very close to a degree bound on the generators
L
of D = k≥0 TSdA (Bk ) when B = A[x0 , x1 , . . . , xr ] is the polynomial ring.
Proposition (3.1.10). Let N be a positive integer and D≤N be the subring of
L
D = k≥0 TSdA (Bk ) generated by elements of degree at most N . Then the inclusion
D≤N ,→ D induces a morphism ψN : Proj(D) → Proj(D≤N ). Further we have
that:
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
19
(i) If B = A[x0 , x1 , . . . , xr ] is a polynomial ring and N ≥ r(d − 1) then ψN is
an isomorphism.
(ii) If A is purely of characteristic zero, i.e., a Q-algebra, then ψN is an isomorphism for any N .
Proof. By Corollary (3.1.9) the morphism ψN is everywhere defined for N ≥ 1. Let
A ,→ A0 be a finite flat extension such that every residue field of A0 has at least d
elements, e.g., A0 = A ⊗Z Λd as in Lemma (1.2.3). If we let C 0 = C ⊗A A0 then
0
we have that D0 = D ⊗A A0 = C 0Sd and D≤N
= D≤N ⊗A A0 as A ,→ A0 is flat.
0
0
0
If ψN : Proj(D ) → Proj(D≤N ) is an isomorphism then so is ψN as A ,→ A0 is
faithfully flat. Replacing A with A0 , it is thus enough to prove the corollary when
every residue field of S has at least
d elements. Hence we can assume that we have
a cover of Proj(D) by D+ f ⊗d with f ∈ B1 by Corollary (3.1.8).
We have that D(f ⊗d ) = TSdA (B(f ) ) and this latter ring is generated by elements
of degree ≤ max{r(d − 1), 1} for arbitrary A and by elements of degree one when
A is purely of characteristic zero by Corollary (1.3.6). As noted in Remark (1.3.3)
this implies that D(f ⊗d ) = (D≤N )(f ⊗d ) which shows (i) and (ii).
Corollary (3.1.11). Let N be a positive integer and DN be the subring of D(N ) =
L
d
d
(N )
induces
k≥0 TSA (BN k ) generated by TSA (BN ). Then the inclusion DN ,→ D
a morphism ψN : Proj(D) → Proj(DN ). Further we have that:
(i) If B = A[x0 , x1 , . . . , xr ] is a polynomial ring and N ≥ r(d − 1) then ψN is
an isomorphism.
(ii) If A is purely of characteristic zero, i.e., a Q-algebra, then ψN is an isomorphism for any N .
L
Proof. Let D≤N be the subring of D = k≥0 TSdA (Bk ) generated by elements of
degree at most N . As Proj(D) is covered in degree one by Corollary (3.1.9) then so
is Proj(D≤N ). In fact, as Proj(D) → Spec(A) is universally closed, it follows that
Proj(D) → Proj(D≤N ) is surjective. By Proposition (2.2.7) (iv) it then follows that
(N )
(N )
DN ,→ (D≤N )
induces an isomorphism Proj(D≤N ) → Proj(DN ). The corollary
thus follows from Proposition (3.1.10).
Theorem (3.1.12). Let S be any scheme and E a quasi-coherent OS -sheaf of finite
type. Then for any N ≥ 1, there is a canonical morphism
Symd (P(E)/S) → P TSdOS (SN E) .
If L is a locally free OS -sheaf of constant
rank r + 1 then the canonical morphism
Symd (P(L)/S) ,→ P TSdOS (SN L) is a closed immersion for N ≥ r(d − 1). In
particular, it follows that Symd (P(L)/S) → S is strongly projective.
Proof. The existence of the morphism follows by Corollary (3.1.11).
Part (i) of the
same corollary shows that Symd (P(L)/S) ,→ P TSdOS (SN L) is a closed immersion
20
DAVID RYDH
when N ≥ r(d − 1). As SN L is locally free of constant rank it follows by paragraph (1.1.7) that TSdOS (SN L) is locally free of constant rank which shows that
Symd (P(L)/S) is strongly projective.
d
In section 3.3,
we will show that the canonical morphism Sym (P(E)/S) →
d
N
P TSOS (S E) is a universal homeomorphism
onto its image which is defined to
be the Chow scheme Chow0,d P(E) .
3.2. The scheme of divided powers. Let S be any scheme and A a quasicoherent sheaf of OS -algebras. As the construction of ΓdA (B) commutes with localization with respect to multiplicatively closed subsets of A we may define a quasi
coherent sheaf of OS -algebras ΓdOS (A). We let Γd (Spec(A)/S) = Spec ΓdOS (A) .
The scheme Γd (X/S) is thus defined for any scheme X affine over S. Similarly we
obtain for any homomorphism of quasi-coherent OS -algebras A → B a morphism
of schemes Γd (Spec(B)/S) → Γd (Spec(A)/S). This defines a covariant functor
X 7→ Γd (X/S) from affine schemes over S to affine schemes over S.
It is more difficult to define Γd (X/S) for any X-scheme S since ΓdA (B) does not
commute with localization with respect to B. In fact, it is not even a B-algebra.
In [I, 3.1] a certain functor ΓdX/S is defined which is represented by Γd (X/S) when
X/S is affine. When X/S is quasi-projective, or more generally an AF-scheme, cf.
Definition (3.1.2), then ΓdX/S is represented by a scheme [I, Thm. 3.1.11]. If X/S
is a separated algebraic space, then ΓdX/S is represented by a separated algebraic
space [I, Thm. 3.4.1].
The object representing ΓdX/S will be denoted by Γd (X/S). We briefly state
some facts about Γd (X/S) used in the other sections. We then show that Γd (X/S)
is (quasi-)projective when X/S is (quasi-)projective.
(3.2.1) The space of divided powers — For any algebraic scheme X separated
above S, there is an algebraic space Γd (X/S) over S with the following properties:
(i) For any morphism S 0 → S, there is a canonical base-change isomorphism
Γd (X/S) ×S S 0 ∼
= Γd (X ×S S 0 /S 0 ).
(ii) If X/S is an AF-scheme, then Γd (X/S) is an AF-scheme.
(iii) If A is a quasi-coherent sheaf on
S such that X = SpecS (A) is affine S,
then Γd (X/S) = SpecS ΓdOS (A) is affine over S.
`n
(iv) If X = i=1 Xi then Γd (X/S) is the disjoint union
a
Γd1 (Xi /S) ×S Γd2 (X2 /S) ×S · · · ×S Γdn (Xn /S).
d1 ,d2 ,...,dn ≥0
d1 +d2 +···+dn =d
(v) If X → S has one of the properties: finite type, finite presentation, locally
of finite type, locally of finite presentation, quasi-compact, finite, integral,
flat; then so has Γd (X/S) → S.
This is Thm. 3.1.11, Prop. 3.1.4, Prop. 3.1.8 and Prop. 4.3.1 of [I].
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
21
(3.2.2) Push-forward of cycles — Let f : X → Y be any morphism of algebraic
schemes separated over S. There is then a natural morphism, push-forward of
cycles, f∗ : Γd (X/S) → Γd (Y /S) which for affine schemes is given by the covariance
of the functor ΓdA (·). If f : X → Y is an immersion (resp. a closed immersion,
resp. an open immersion) then f∗ : Γd (X/S) → Γd (Y /S) is an immersion (resp.
closed immersion, resp. open immersion) [I, Prop. 3.1.7].
(3.2.3) Addition of cycles — Let d, e be positive integers. The composition of
the open and closed immersion Γd (X/S) ×S Γe (X/S) ,→ Γd+e (X q X/S) given
by (3.2.1) (iv) and the push-forward Γd+e (X q X/S) → Γd+e (X/S) along the
canonical morphism X q X → X is called addition of cycles [I, Def. 4.1.1].
(3.2.4) The Sym-Gamma morphism — Let X/S be a separated algebraic space
and let (X/S)d = X ×S X ×S · · · ×S X. There is an integral surjective morphism
ΨX : (X/S)d → Γd (X/S), given by addition of cycles, invariant under the permutation of the factors. This gives a factorization (X/S)d → Symd (X/S) → Γd (X/S)
and we denote the second morphism by SGX [I, Prop. 4.1.5].
(3.2.5) Local description of the scheme of divided powers — If U ⊆ X is an open
subset, then we have already seen that Γd (U/S) ⊆ Γd (X/S) is an open subset.
Moreover, there is a cartesian diagram
/ Symd (U/S)
(U/S)d
(X/S)d
/ Symd (X/S)
SGU
/ Γd (U/S)
/ Γd (X/S).
S
S
If X/S is an AF-scheme, then there are
Zariski-covers S = Sα and X = Uα of
S
affine schemes such that Γd (X/S) = Γd (Uα /Sα ). This gives a local description
of the SG-map. For an arbitrary separated algebraic space X/S there is a similar
étale-local description of SG. If U → X is an étale morphism, then there is a
cartesian diagram
/ Symd (U/S)|fpr
(U/S)d |fpr
(X/S)d
SGX
/ Symd (X/S)
/ Γd (U/S)|reg
/ Γd (X/S)
where the vertical arrows are étale [I, Prop. 4.2.4]. Here fpr and reg denotes the
open
maps are fixed-point reflecting and regular. If
` locus where the corresponding`
Uα → X is an étale cover, then
Γd (Uα /S)|fpr → Γd (X/S) is an étale cover.
Thus, we have an étale-local description of SG in affine schemes.
We now give a similar treatment of Γd (X/S) for X = Proj(B) projective as that
given for Symd (X/S) in the previous section.
22
DAVID RYDH
Proposition (3.2.6). Let S = Spec(A) where A is affine and let X = Proj(B)
where B is a graded A-algebra finitely generated in degree one. Then Γ(X/S) is
covered by open subsets of the form Γd (Xf /S) with f ∈ Bn for some n. If every
residue field of S has at least d elements, then is enough to consider open subsets
with n = 1.
Proof. Follows from Proposition (3.1.7) using that ΨX ((Xf /S)d ) = Γd (Xf /S). Proposition (3.2.7). Let A be L
a ring and B a graded A-algebra finitely generated
d
in degree one. Let W = Proj
k≥0 ΓA (Bk ) . Then W is covered by the open
subsets of the form Wγ d (b) where b ∈ Bn for some n. If every residue field of
S = Spec(A) has at least d elements, then is enough to consider open subsets with
b ∈ B1 .
Proof. Let F be a graded flat A-algebra
with a surjection
F B. Consider the
L
L
d
d
Γ
(F
)
Γ
induced surjective homomorphism
k
k≥0 A (Bk ) and the correk≥0 A L
0
d
sponding closed immersion W ,→ W = Proj
k≥0 ΓA (Fk ) . The open subset of
d
0
d
W = Sym (Proj(F )/S) given by γ (f ) = 0, where f ∈ Fn , coincides with the
open subset Symd (Proj(F )f /S). These subsets cover W 0 by Corollary (3.1.8). The
proposition follows immediately.
Corollary (3.2.8). Let S be any scheme and let A be a graded quasi-coherent
OS -algebra of finitetype generated in degree one. Then Γd Proj(A)/S and W =
L
d
Proj
k≥0 ΓA (Ak ) are canonically isomorphic. Under this isomorphism, the open
d
subset Γ Proj(A)f is identified with Wγ d (f ) for any homogeneous element f ∈ A.
Proof. By Propositions (3.2.6) and (3.2.7) the open subsets Γd Spec A(f ) and
Wγ d (f ) for f ∈ An , covers Γd (Proj(A)) and W respectively. As these subsets are
canonically isomorphic the corollary follows.
Proposition (3.2.9). Let S be any scheme and let A be L
a graded quasi-coherent
OS -algebra of finite type generated in degree one. Let D = k≥0 ΓdA (Ak ). Let N be
L
a positive integer and let DN be the subring of D(N ) = k≥0 ΓdA (AN k ) generated
by ΓdA (AN ). The inclusion DN ,→ D(N ) induces a morphism ψN : Proj(D) →
Proj(DN ). Furthermore
(i) If A is locally generated by at most r + 1 elements and N ≥ r(d − 1) then
ψN is an isomorphism.
(ii) If S is purely of characteristic zero, i.e., a Q-scheme, then ψN is an isomorphism for every N .
Proof. The statements are local on S so we may assume that S = Spec(A) is
e where B is a graded A-algebra finitely generated in degree one.
affine and A = B
L
d
0
Choose a surjection B 0 = A[x0 , x1 , . . . , xr ] B. Let D =
k≥0 ΓA (Bk ), D =
L
d
0
0
(N )
and D0(N ) generated
k≥0 ΓA (Bk ) and let DN and DN be the subrings of D
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
23
by degree one elements. Then we have a commutative diagram
/ / DN
_
0
DN
_
(3.2.9.1)
D0
(N )
◦
/ / D(N ) .
(N )
0
induces a morphism
,→ D0
By Corollary (3.1.11) the inclusion DN
0
0
ψN
: Proj D0(N ) → Proj(DN
)
having properties (i) and (ii). From the commutative diagram (3.2.9.1) it follows
that the inclusion DN ,→ D(N ) induces a morphism ψN : Proj(D(N ) ) → Proj(DN )
with the same properties.
Theorem (3.2.10). If X → S is strongly projective (resp. strongly quasi-projective)
then Γd (X/S) → S is strongly projective (resp. strongly quasi-projective). If X → S
is projective (resp. quasi-projective) and S is quasi-compact and quasi-separated
then Γd (X/S) → S is projective (resp. quasi-projective).
Proof. In the strongly projective (resp. strongly quasi-projective) case we immediately reduce to the case where X = PS (L) for some locally free OS -module L
of finite rank r + 1, using the push-forward (3.2.2), and the result follows from
Theorem (3.1.12).
If S is quasi-compact and quasi-separated and X → S is projective (resp. quasiprojective) then there is a closed immersion (resp. immersion) X ,→ PS (E) for some
quasi-coherent OS -module E of finite
It is enough to show that Γ(PS (E)) is
L type.
d k
projective. As Γ(PS (E)) = Proj
k≥0 Γ (S (E) by Corollary (3.2.8), this follows
from Proposition (3.2.9) and the quasi-compactness of S.
3.3. The Chow scheme. Let k be a field and let E be a vector space over k with
basis x0 , x1 , . . . , xn . Let E ∨ be the dual vector space with dual basis y0 , y1 , . . . , yn .
Let X = P(E) = Pnk . If k 0 /k is a field extension then a point x : Spec(k 0 ) → X
0
is given by coordinates
Pn(x0 : x1 : ·0· · : xn ) in k . To x we associate the Chow form
Fx (y0 , y1 , . . . , yn ) = i=0 xi yi ∈ k [y0 , y1 , . . . , yn ] which is defined up to a constant.
A zero-cycle on X = Pnk is a formal sum of closed points. To any zero-dimensional
subscheme Z ,→ X we associate
the zero-cycle [Z] defined as the sum of its points
P
with multiplicities. If Z = j aj [zj ] is a zero-cycle on X and k 0 /k a field extension
P
0
then we let Zk0 = Z ×k k 0 =
j aj [zj ×k k ]. It is clear that if Z ,→ X is a
0
zero-dimensional subscheme then [Z] ×k k = [Z ×k k 0 ].
We say that
are positive. The degree of a
P a cycle is effective if its coefficients
P
cycle Z = j aj [zj ] is defined as deg(Z) = j aj deg k(zj )/k . It is clear that
deg(Zk0 ) = deg(Z) for any field extension k 0 /k.
Let Z
on X and choose a field extension k 0 /k such that
Pbe an 0effective zero-cycle
0
Zk0 = j aj [zj ] is a sum of k -points, i.e., k(zj0 ) = k 0 . We then define its Chow
Q a
form as FZ = j Fz0j . It is easily seen that
j
24
DAVID RYDH
(i) FZ does not depend on the choice of field extension k 0 /k.
(ii) FZ has coefficients in k.
(iii) The degree of FZ coincides with the degree of Z.
(iv) Z is determined by FZ .
Further, if k is perfect there is a correspondence between zero-cycles of degree d on
X and Chow forms of degree d, i.e., homogeneous polynomials, F ∈ k[y0 , y1 , . . . , yn ]
which splits into d linear forms after a field extension. The Chow
forms of degree d
with coefficients in k is a subset of the linear forms on P Sd (E ∨ ) and thus a subset
of the k-points of P Sd (E ∨ )∨ = P TSd (E) .
(3.3.1) The Chow variety — Classically it is shown that for r ≥ 0 and d ≥ 1
there is a closed subset of P Tr+1 (TSd (E)) parameterizing r-cycles of degree d
on P(E). The Chow variety Chowr,d P(E) is then taken as the reduced scheme
corresponding to this subset. More generally, if S is any scheme
and L is a locally
free sheaf then there is a closed subset of PS Tr+1 (TSd (L)) parameterizing r-cycles
of degree d on PS (L).
We will now show that the classical Chow variety parameterizing zero-cycles of
degree d has a canonical closed subscheme structure. We begin with the case where
S is the spectrum of a field.
(3.3.2) The Chow scheme for P(E)/k — Let k 0 /k be a field extension such that k 0
is algebraically closed. As (P(E)/k)d → Symd (P(E)/k) is integral, it is easily seen
that a k 0 -point of Symd (P(E)/k) corresponds to an unordered tuple (x1 , x2 , . . . , xd )
of k 0 -points of P(E). Assigning such a tuple the
Chow form of the cycle [x1 ] + [x2 ] +
· · · + [xd ] gives a map Hom k 0 , Symd (P(E)/k) → Hom k 0 , P(TSd (E)) . It is easily
seen to be compatible with the homomorphism of algebras
M
M
Sk TSd (E) →
TSd Sk (E)
k≥0
k≥0
and thus extends to a morphism of schemes
Symd P(E)/k → P TSd (E) .
It is further clear that the image of this morphism
consists of the Chow forms of
degree d and that Symd P(E)/k → P TSd (E) is universally injective and hence
a universal homeomorphism
onto its image as Symd P(E)/k is projective. We let
Chow0,d P(E) be the scheme-theoretical image of this morphism.
More generally, we define Chow0,d P(L)/S for any locally free sheaf L on S as
follows:
Definition-Proposition (3.3.3). Let S be a scheme and L a locally free OS -sheaf
L
L
d
d
k
k
of finite type. Then the homomorphism
k≥0 S TSOS (L) →
k≥0 TSOS (S L)
induces a morphism
ϕL : Symd P(L)/S → P TSdOS (L)
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
25
which is a universal homeomorphism onto its image. We let Chow0,d P(L) be its
scheme-theoretic image.
f
Proof. The question is local so we can assume that S = Spec(A) and L = M
where M is a free A-module of finite rank. Corollary (3.1.11), with N = 1 and
L
L
L
B = k≥0 Sk M , shows that k≥0 Sk TSdA (M ) → k≥0 TSdA (Sk M ) induces a well
defined morphism Symd P(L)/S → P TSdOS (L) .
To show that Symd P(L)/S → P TSdOS (L) is a universal homeomorphism onto
its image it is enough to show that it is universally injective as Symd P(L)/S → S
is universally closed. As L is flat over S the symmetricproduct commutes
with base
change and it is enough to show that Symd P(L)/S → P TSdOS (L) is injective
when S is a field. This was discussed above.
If X ,→ P(L)
is a closed immersion (resp. an immersion) then the subset of
Chow0,d P(L) parameterizing cycles with support in X is closed (resp. locally
closed). In fact, it is the image of the morphism
(3.3.3.1)
Symd (X/S) → Symd P(L)/S → Chow0,d P(L) .
As Symd P(L)/S = Γd P(L)/S , this morphism factors through Symd (X/S) →
Γd (X/S). Moreover, as Symd (X/S) → Γd (X/S) is a homeomorphism [I, Cor. 4.2.5],
the morphism
(3.3.3.2)
Γd (X/S) → Symd P(L)/S → Chow0,d P(L) .
has the same image as (3.3.3.1). Since Γd is more well-behaved, e.g., commutes
with base change S 0 → S, the following definition is reasonable:
Definition (3.3.4). Let S be any scheme and L a locally
free sheaf on S. If X ,→
X
,→
P(L)
be the scheme-theoretic
P(L) is a closed immersion we let
Chow
0,d
d
d
image of Γ (X/S) ,→ Γ P(L)/S → Chow0,d P(L) . If X ,→ P(L) is an immer
sion we let Chow0,d X ,→ P(L) be the open subscheme of Chow0,d X ,→ P(L)
corresponding to cycles with support in X.
Remark (3.3.5). Classically Chow0,d X ,→ P(L) is defined as the reduced sub
scheme of Chow0,d P(L) ,→ P TSd (L) parameterizing zero-cycles of degree d with
support
in X. It is clear that this is the reduction of the scheme Chow0,d X ,→
P(L) as defined in Definition (3.3.4).
Remark (3.3.6).
If L is a locally free sheaf on S of finite type then by definition
Chow0,d P(L) is Proj(B) where B is the image of
M
M
Sk TSd (L) →
TSd Sk (L)
k≥0
k≥0
i.e., B is the subalgebra of k≥0 TSd Sk (L) generated by degree one elements. If
X ,→ P(L) is a closed immersion then X = Proj(A) where A is a quotient of S(L).
L
26
DAVID RYDH
The Chow scheme Chow0,d X ,→ P(L) is then Proj(B) where B is the subalgebra
L
of k≥0 ΓdOS (Ak ) generated by degree one elements, cf. Corollary (3.2.8).
Proposition (3.3.7). Let S be any scheme and let B be a graded quasi-coherent
OS -algebra of finite type generated in degree one. Then there is a canonical morphism
ϕB : Γd Proj(B)/S → P ΓdOS (B1 )
which is a universal homeomorphism onto its image. This morphism commutes
with base change S 0 → S and surjections B B 0 .
Proof. The existence of the morphism follows from Proposition (3.2.9). That ϕB is
universally injective can be checked on the fibers and this is done in the beginning
of this section. The last statements follows from the corresponding statements of
the algebra of divided powers.
Remark (3.3.6) and Proposition (3.3.7)
shows that there is a natural extension
of the definition of Chow0,d X ,→ PS (L) which includes the case where L need not
be locally free. In particular, we obtain a definition valid for arbitrary projective
schemes:
Definition (3.3.8). Let X/S be quasi-projective morphism of schemes and let
X ,→ PS (E) be an immersion for some quasi-coherent OS -module E of finite type.
Let X be the scheme-theoretic image of X in PS (E) which can be written as X =
Proj(B) where B is a quotient of S(E). We let Chow0,d X ,→ PS (E) be the
scheme-theoretic image of ϕB : Γd Proj(B)/S → P ΓdOS (B1 ) or equivalently, the
scheme-theoretic image of
ϕX,E : Γd Proj(B)/S → P ΓdOS (B1 ) ,→ P ΓdOS (E) .
We let Chow0,d X ,→ PS (E) be the open subscheme of Chow0,d X ,→ PS (E)
given by the image of
Γd (X/S) ⊆ Γd (X/S) → Chow0,d X ,→ PS (E) .
This is indeed an open subscheme as Γd X/S → Chow0,d X ,→ P(E) is a homeomorphism by Corollary (3.3.7).
Remark (3.3.9). Let S be any scheme, E a quasi-coherent OS -module and X ,→
P(E) an immersion. Let S 0 → S be any morphism and let X 0 = X ×S S 0 and
E 0 = E ⊗OS OS 0 . There is a commutative diagram
Γd (X 0 /S 0 )
∼
=
Γd (X/S) ×S S 0
ϕX 0 ,E 0
◦
ϕX,E ×S idS 0
/ PS 0 (E 0 )
∼
=
/ PS (E) ×S S 0
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
27
i.e., ϕX 0 ,E 0 = ϕX,E ×S idS 0 . As the underlying sets of Chow0,d X ,→ P(E) and
Chow0,d X 0 ,→ P(E 0 ) are the images of ϕX,E and ϕX 0 ,E 0 it follows that the canonical morphism
(3.3.9.1)
Chow0,d X 0 ,→ PS 0 (E 0 ) ,→ Chow0,d X ,→ PS (E) ×S S 0
is a nil-immersion, i.e., a bijective closed immersion. As the scheme-theoretic image
commutes with flat base change [EGAIV , Lem. 2.3.1] the morphism (3.3.9.1) is an
isomorphism if S 0 → S is flat.
If Z ,→ X is an immersion (resp. a closed immersion, resp. an open immersion)
then there is an immersion (resp. a closed immersion, resp. an open immersion)
Chow0,d Z ,→ PS (E) ,→ Chow0,d X ,→ PS (E) .
Proposition (3.3.10). Let S = Spec(A) where A is affine and such that every
residue field of S has at least d elements. Let X =L
Proj(B) where B is a graded
A-algebra finitely generated in degree one. Let D = k≥0 ΓdA Bk ) and let E ,→ D
be the subalgebra generated by elements of degree one. Then Chow X ,→ P(B1 ) =
Proj(E) is covered by open subsets of the form Spec(Eγ d (f ) ) with f ∈ B1 . Fur
thermore, Eγ d (f ) is the subalgebra of Γd B(f ) generated by elements of degree one,
i.e., elements of the form ×ni=1 γ di (bi /f ) with bi ∈ B1 .
Proof. The first statement follows immediately from Proposition (3.2.7) taking into
account that the inclusion E ,→ D induces a surjective morphism Proj(D) →
Proj(E) by Proposition (3.3.7). The last statement is obvious.
4. The relations between the parameter spaces
In this section we show that the morphisms
Symd (X/S) → Γd (X/S) → Chow0,d X ,→ P(E)
are universal homeomorphisms with trivial residue field extensions. That the
first morphism is a universal homeomorphisms with trivial residue field extensions is shown in [I]. We also briefly mention the construction of the morphism
Hilbd (X/S) → Symd (X/S).
4.1. The Sym-Gamma morphism. In this section we discuss some properties of
the canonical morphism SGX : Symd (X/S) → Γd (X/S) defined in (3.2.4). Recall
the following basic result:
Proposition (4.1.1). [I, Cor. 4.2.5] Let X/S be a separated algebraic space. The
canonical morphism SGX : Symd (X/S) → Γd (X/S) is a universal homeomorphism
with trivial residue field extensions. If S is purely of characteristic zero or X/S is
flat, then SGX is an isomorphism.
From Proposition (4.1.1) we obtain the following results which only concerns
Symd (X/S) but relies on the existence of the well-behaved functor Γd and the
morphism Symd (X/S) → Γd (X/S).
28
DAVID RYDH
Corollary (4.1.2). Let S → S 0 be a morphism of schemes and X/S a separated
algebraic space. The induced morphism Symd (X 0 /S 0 ) → Symd (X/S) ×S S 0 is a
universal homeomorphism with trivial residue field extensions. If S 0 is of characteristic zero then this morphism is an isomorphism. If X 0 /S 0 is flat then the
morphism is a nil-immersion.
Proof. Follows from Proposition (4.1.1) and the commutative diagram
/ Symd (X/S) ×S S 0
Symd (X 0 /S 0 )
Γd (X 0 /S 0 )
◦
/ Γd (X/S) ×S S 0 .
∼
=
Corollary (4.1.3). Let X/S be a separated algebraic space and Z ,→ X a closed
subscheme. Let q : (X/S)d → Symd (X/S)
be the quotient morphism. The induced
morphism Symd (Z/S) → q (Z/S)d is a universal homeomorphism with trivial
residue field extensions. If S is of characteristic zero then this morphism is an
isomorphism. If Z/S is flat then the morphism is a nil-immersion.
Proof. Follows from Proposition (4.1.1) and the commutative diagram
/ Symd (X/S)
Symd (Z/S)

Γd (Z/S) ◦
/ Γd (X/S).
Let us also mention the following result.
Theorem (4.1.4). Let A be any ring, let B be an A-algebra and let d be a positive
integer. Let ϕ : ΓdA (B) → TSdA (B) be the canonical homomorphism. Then
(i) If x ∈ ker(ϕ) then d!x = 0 and xd! = 0.
(ii) If y ∈ TSdA (B) then d!y ∈ im(ϕ) and y d! ∈ im(ϕ).
Proof. This is (1.1.9) and [II, Cor. 4.7].
It is not difficult to prove that if every prime but p is invertible in A, then d! in
the theorem can be replaced with the highest power of p dividing d!.
Examples (4.1.5). The following examples are due to C. Lundkvist [Lun08]:
(i) An A-algebra B such
(ii) An A-algebra B such
(iii) A surjection B → C
surjective
(iv) An A-algebra B such
that ΓdA (B) → TSdA (B) is not injective
that ΓdA (B) → TSdA (B) is not surjective
of A-algebras such that TSdA (B) → TSdA (C) is not
that ΓdA (B)red ,→ TSdA (B)red is not an isomorphism
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
29
(v) An A-algebra B and a base change A → A0 such that the canonical homomorphism TSdA (B) ⊗A A0 → TSdA0 (B 0 ) is not injective.
(vi) An A-algebra B and a base change A → A0 such that the canonical homomorphism TSdA (B) ⊗A A0 → TSdA0 (B 0 ) is not surjective.
Remark (4.1.6). The seminormalization of a scheme X is a universal homemomorphism with trivial residue fields X sn → X such that any universal homeomorphism
with trivial residue field X 0 → X factors uniquely through X sn → X [Swa80]. If
X sn = X then we say that X is seminormal. If X → Y is a morphism and X is
seminormal then X → Y factors canonically through Y sn → Y .
Using Proposition (4.1.1) it can be shown that Symd (X/S)sn = Symd (X sn /S sn )sn .
Corollaries (4.1.2) and (4.1.3) then show that in the fibered category of seminormal
schemes Sch sn , taking symmetric products commutes with arbitrary base change
and closed subschemes. This is a special property for Symd which does not hold
for arbitrary quotients.
4.2. The Gamma-Chow morphism. Let us first restate the contents
of Proposition (3.2.9) taking into account the definition of Chow0,d X ,→ P(E) .
Proposition (4.2.1). Let S be a scheme, q : X → S quasi-projective and E a
quasi-coherent OS -module of finite type such that there is an immersion X ,→ P(E).
Let k ≥ 1 be an integer. Then
(i) The canonical map
M
S ΓdOS (Sk E) →
ΓdOS (Ski E)
i≥0
induces a morphism
ϕE,k : Γd (X/S) ,→ Γd P(E)/S → P ΓdOS (Sk E)
which is a universal homeomorphism onto its image. The
scheme-theoretical
image of ϕE,k is by definition Chow0,d X ,→ P(E ⊗k ) .
(ii) Assume that either E is locally generated by at most r + 1 elements and
k ≥ r(d − 1) or S has pure characteristic zero, i.e., is a Q-scheme. Then
ϕE,k is a closed immersion and Γd (X/S) → Chow0,d X ,→ P(E ⊗k ) is an
isomorphism.
Remark (4.2.2). As Γd (X/S) → Chow0,d X ,→ P(E) is a universal homeomorphism, the topology of the Chow scheme does not depend on the chosen embedding
X ,→ P(E).
In higher dimension, it is well-known that the Chow variety Chowr,d X ,→ P(E)
does not depend on the embedding X ,→ P(E) as a set. This follows from the fact
that a geometric point corresponds to an r-cycle of degree d [Sam55, §9.4d,h]. The
invariance of the topology is also well-known, cf. [Sam55, §9.7]. This implies that
the weak normalization of the Chow variety does not depend on the embedding in
the analytic case, cf. [AN67]. This also follows from functorial descriptions of the
30
DAVID RYDH
Chow variety over weakly normal schemes as in [Gue96] over C or more generally
in [Kol96, §1.3]. We will now show that the residue fields of Chow0,d X ,→ P(E)
do not depend on the embedding.
Proposition (4.2.3). Let S, q : X → S and P(E)
as in Proposition (4.2.1). The
morphism ϕE : Γd (X/S) → Chow0,d X ,→ P(E) is a universal homeomorphism
with trivial residue field extensions.
Proof. We have already seen that the morphism ϕE is a universal homeomorphism.
It is thus enough to show that it has trivial residue field extensions. To show
this
it is enough to show that for every point a : Spec(k) → Chow0,d X ,→ P(E) with
k = k sep there exists a, necessarily unique, point b : Spec(k) → Γd (X/S) lifting a,
i.e., the diagram
ϕE
/ Chow0,d X ,→ P(E)
Γd (X/S)
h
O
a
b
Spec(k)
d
has a unique filling. By
(3.2.1) (i) and Remark (3.3.9) the schemes Γ (X/S) and
Chow0,d (X ,→ P(E)) red commute with base change, i.e.,
Γd (X/S) ×S S 0 = Γd (X ×S S 0 /S 0 )
Chow0,d (X ,→ P(E)) ×S S 0 red = Chow0,d X ×S S 0 ,→ P(E ⊗OS OS 0 ) red
for any S 0 → S. We can thus assume that S = Spec(k) and hence that the image
of a is a closed point.
Let r + 1 be the rank of E. The point a then corresponds to a Chow form
−∞
Fa ∈ k[y0 , y1 , . . . , yr ] which is homogeneous of degree d. Over k = k p
this form
factors into linear forms
Fa = F1d1 F2d2 . . . Fndn
P (j)
(j)
(j) where d = d1 + d2 + · · · + dn . Let Fj = j xi yi and let k x(j) = k x0 , . . . , kr .
pe
If we let d = pe m such that p - m, then k x(j)
⊆ k as Fidi is k-rational. Thus the
(j)
e
/k is at most p and it follows by [II, Prop. 7.6] that Γdj (X/S)
exponent of k x
has a unique k-point bj corresponding to Fj . The k-point b = b1 + b2 + · · · + bn is
a lifting of a.
Remark (4.2.4). Proposition (4.2.3) also follows from the following fact. Let k be
a field, E a k-vector space and X ,→ Pk (E) a subscheme. Let Z be an r-cycle
on X. The residue field of the point corresponding to Z in the Chow variety
Chowr,d X ,→ P(E) , the Chow field of Z, does not depend on the embedding
X ,→ Pnk [Kol96, Prop-Def. I.4.4].
As ϕE ⊗k : Γd (X/S) → Chow0,d X ,→ P(E ⊗k ) is an isomorphism for sufficiently
large k by Proposition (4.2.1) the Chow field coincides with the corresponding
residue field of Γd (X/S).
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
31
4.3. The Hilb-Sym morphism. The Hilbert-Chow morphism can be constructed
in several different ways. Mumford [GIT, Ch. 5 §4] constructs a morphism
Hilbd (Pn ) → Divd ((Pn )∨ ) ∼
= P(O(Pn )∨ (d))
from the Hilbert scheme of d points to Cartier divisors of degree d on the dual
space. This construction is a generalization of the construction of the Chow variety
and it follows immediately that the image of this morphism is the Chow variety of
Pn . As the Chow variety Chow0,d (Pn ) does not coincide with Symd (Pn ) = Γd (Pn )
in general, it is not clear that this construction lifts to a morphism to Symd (Pn )
or to Γd (Pn ). Neeman solved this [Nee91] constructing a morphism Hilbd (Pn ) →
Symd (Pn ) directly without using Chow forms.
There is a natural way of constructing the “Hilbert-Chow” morphism due to
Grothendieck [FGA] and Deligne [Del73]. There is a natural map Hilbd (X/S) →
Γd (X/S) taking a flat family Z → T to its norm family [II]. We will call this
map the Grothendieck-Deligne norm map and denote it with HGX . Using that the
symmetric product coincide with the space of divided powers for X/S flat, it follows
by functoriality that the morphism Hilbd (X/S) → Γd (X/S) factors through the
symmetric product. To be precise, we have the following natural transformation:
Definition (4.3.1). Let X/S be a separated algebraic space. We let HSX :
Hilbd (X/S) → Symd (X/S) be the following morphism. Let T be an S-scheme and
f : T → Hilbd (X/S) a T -point. Then f corresponds to a subscheme Z ,→ X ×S T
which is flat and finite over T . There is a commutative diagram
/ Hilbd (Z/T )
T LL
_
LLL
LLL
L
(f,idT ) LLL
&
d
Hilb (X/S) ×S T
HGZ
HGX
/ Γd (Z/T ) o
_
SGZ
∼
=
Symd (Z/T )
/ Γd (X/S) ×S T o
SGX
Sym (X/S) ×S T
d
and we let HSX (f ) be the composition T → Symd (X/S) ×S T → Symd (X/S).
The morphisms HSX and HGX are isomorphisms when X/S is a smooth curve [I].
They are also both isomorphisms over the non-degeneracy locus as shown in the
next section. In [ES04, RS07], it is shown that the closure of the non-degeneracy
locus of Hilbd (X/S) — the good component — is a blow-up of either Γd (X/S) or
Symd (X/S).
5. Outside the degeneracy locus
In this section we will prove that the morphisms
Hilbd (X/S) → Symd (X/S) → Γd (X/S) → Chow0,d X ,→ P(E)
32
DAVID RYDH
are all isomorphisms over the open subset parameterizing “non-degenerated families” of points. That the morphism HGX : Hilbd (X/S) → Γd (X/S) is an isomorphism over the non-degeneracy locus is shown in [II]. It is thus enough to show
that the last two morphisms are isomorphisms outside the degeneracy locus.
5.1. Families of cycles. Let k be an algebraically closed field and fix a geometric
point s : Spec(k) → S. Let α : Spec(k) → Symd (X/S) be a geometric point
above s. As (X/S)d → Symd (X/S) is integral, we have that α lifts (non-uniquely)
to a geometric point β : Spec(k) → (X/S)d . Let πi : (X/S)d → X be the ith
projection and let xi = πi ◦ β. It is easily seen that the different liftings β of α
corresponds to the permutations of the d geometric points xi : Spec(k) → X. This
gives a correspondence between k-points of Symd (X/S) and effective zero-cycles of
degree d on Xs .
As Symd (X/S) → Γd (X/S) → Chow0,d X ,→ P(E) are universal homeomorphisms, there is a bijection between their geometric points.
It is thus reasonable
to say that Symd (X/S), Γd (X/S) and Chow0,d X ,→ P(E) parameterize effective
zero-cycles
of degree d. Moreover, as Symd (X/S) → Γd (X/S) → Chow0,d X ,→
P(E) have trivial residue field extensions, there is a bijection between k-points for
any field k.
Definition (5.1.1). Let X, S, k and s be as above. Let Z be an effective zerocycle of degree d on Xs . The residue
field of the corresponding point in Symd (X/S),
Γd (X/S) or Chow0,d X ,→ P(E) is called the Chow field of Z.
Definition (5.1.2). Let k be a field and X a scheme over k. Let k 0 /k and k 00 /k be
field extensions of k. Two cycles Z 0 and Z 00 on X ×k k 0 and X ×k k 00 respectively,
are said to be equivalent if there is a common field extension K/k of k 0 and k 00 such
that Z 0 ×k0 K = Z 00 ×k00 K. If Z 0 is a cycle on X ×S k 0 equivalent to a cycle on
X ×S k 00 then we say that Z 0 is defined over k 00 .
Remark (5.1.3). If Z is a cycle on X ×S k then the corresponding morphism
Spec(k) → Symd (X/S) factors through Spec(k) → Spec(k). Thus if Z is defined over a field K then the Chow field is contained in K. Conversely it can be
shown that Z is defined over an inseparable extension of the Chow field. Thus,
in characteristic zero the Chow field of Z is the unique minimal field of definition
of Z. In positive characteristic, it can be shown that the Chow field of Z is the
intersection of all minimal fields of definitions of Z, cf. [Kol96, Thm. I.4.5] and [II,
Prop. 7.13].
Let T be any scheme
and f : T → Symd (X/S), f : T → Γd (X/S) or f : T →
Chow0,d X ,→ P(E) a morphism. A geometric k-point of T then corresponds to a
zero-cycle of degree d on X ×S k. The following definition is therefore natural.
Definition (5.1.4). A family of cycles parameterized
by T is a T -point of either
Symd (X/S), Γd (X/S) or Chow0,d X ,→ P(E) . We use the notation Z → T to
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
33
denote a family of cycles parameterized by T and let Zt be the cycle over t, i.e.,
the cycle corresponding to k(t) → T → Symd (X/S), etc.
As Γd (X/S) commutes with base change and has other good properties it is the
“correct” parameter scheme and the morphisms T → Γd (X/S) are the “correct”
families of cycles.
5.2. Non-degenerated families.
(5.2.1) Non-degenerate families of subschemes — Let k be a field and X be a
k-scheme. If Z ,→ X is a closed subscheme then it is natural say that Z is nondegenerate if Zk is reduced, i.e., if Z → k is geometrically reduced. If Z is of
dimension zero then Z is non-degenerate if and only if Z → k is étale. Similarly
for any scheme S, a finite flat morphism Z → S of finite presentation is called a
non-degenerate family if every fiber is non-degenerate, or equivalently, if Z → S is
étale.
Let Z → S be a family of zero dimensional subschemes, i.e., a finite flat morphism
of finite presentation. The subset of S consisting of points s ∈ S such that the fiber
Zs → k(s) is non-degenerate is open [EGAIV , Thm. 12.2.1 (viii)]. Thus, there is an
open subset Hilbd (X/S)nd of Hilbd (X/S) parameterizing non-degenerate families.
P
(5.2.2) Non-degenerate families of cycles — A zero-cycle Z =
i ai [zi ] on a
k-scheme X is called non-degenerate if every point in the support of Zk has multiplicity one. Equivalently the multiplicities ai are all one and the field extensions
k(zi )/k are separable. It is clear that there is a one-to-one correspondence between
non-degenerate zero-cycles on X and non-degenerated zero-dimensional subschemes
of X.
Given a family of cycles Z → S, i.e., a morphism S → Symd (X/S), S →
d
Γ (X/S) or S → Chow0,d X ,→ P(E) , we say that it is non-degenerate family if
Zs is non-degenerate for every s ∈ S.
(5.2.3) Degeneracy locus of cycles — Let X → S be a morphism of schemes
and let ∆ ,→ (X/S)d be the big diagonal, i.e., the union of all diagonals ∆ij :
(X/S)d−1 → (X/S)d . It is clear that the image of ∆ by (X/S)d → Symd (X/S)
parameterizes degenerate cycles and that the open complement parameterizes non
degenerate cycles. We let Symd (X/S)nd , Γd (X/S)nd and Chow0,d X ,→ P(E) nd
be the open subschemes of Symd (X/S), Γd (X/S) and Chow0,d X ,→ P(E) respectively, parameterizing non-degenerate cycles.
We will now give an explicit cover
of the degeneracy locus of Symd (X/S),
Γ (X/S) and Chow0,d X ,→ P(E) . Some of the notation is inspired by [ES04,
2.4 and 4.1] and [RS07].
d
Definition (5.2.4). Let A be a ring and B an A-algebra. Let x = (x1 , x2 , . . . , xd ) ∈
B d . We define the symmetrization and anti-symmetrization operators from B d to
34
DAVID RYDH
TdA (B) as follows
s(x) =
X
xσ(1) ⊗A xσ(2) ⊗A · · · ⊗A xσ(d)
σ∈Sd
a(x) =
X
(−1)|σ| xσ(1) ⊗A xσ(2) ⊗A · · · ⊗A xσ(d) .
σ∈Sd
As s and a are A-multilinear, s is symmetric and a is alternating it follows that we
get induced homomorphisms, also denoted s and a
s : SdA (B) → TSdA (B)
Vd
a : A (B) → TdA (B).
Remark (5.2.5). If d is invertible in A, then sometimes the symmetrization and
1
1
s and d!
a. We will never use this
anti-symmetrization operators are defined as d!
convention. In [ES04] the tensor a(x) is denoted by ν(x) and referred to as a norm
vector.
Definition (5.2.6). Let A be a ring and B an A-algebra. Let x = (x1 , x2 , . . . , xd ) ∈
B d and y = (y1 , y2 , . . . , yd ) ∈ B d . We define the following element in ΓdA (B)
δ(x, y) = det γ 1 (xi yj ) × γ d−1 (1) ij .
Following [RS07] we call the ideal I = IA = δ(x, y) x,y∈B d , the canonical ideal.
As δ is multilinear and alternating in both arguments we extend the definition of δ
to a function
Vd
Vd
Vd
δ : A (B) × A (B) → S2A A (B) → ΓdA (B).
Proposition (5.2.7) ([ES04, Prop. 4.4]). Let A be a ring, B an A-algebra and
x, y ∈ B d . The image of δ(x, y) by ΓdA (B) → TSdA (B) ,→ TdA (B) is a(x)a(y). In
particular, a(x)a(y) is symmetric.
Lemma (5.2.8) ([ES04, Lem. 2.5]). Let A be a ring and let B and A0 be A-algebras.
Let B 0 = B ⊗A A0 . Denote by IA ⊂ ΓdA (B) and IA0 ⊂ ΓdA0 (B 0 ) = ΓdA (B) ⊗A A0 the
canonical ideals corresponding to B and B 0 . Then IA A0 = IA0 .
Lemma (5.2.9). Let S be a scheme and X and S 0 be S-schemes. Let X 0 = X ×S S 0 .
Let ϕ : Γd (X 0 /S 0 ) = Γd (X/S) ×S S 0 → Γd (X/S) be the projection morphism. The
inverse image by ϕ of the degeneracy locus of Γd (X/S) is the degeneracy locus of
Γd (X 0 /S 0 ).
Proof. Obvious as we know that a geometric point Spec(k) → Γd (X/S) corresponds
to a zero-cycle of degree d on X ×S Spec(k).
Lemma (5.2.10). Let k be a field and let B be a k-algebra generated as an algebra
by the k-vector field V ⊆ B. Let k 0 /k be a field extension
and let x1 , x2 , . . . , xd be
d distinct k 0 -points of Spec(B ⊗k k 0 ). If k has at least d2 elements then there is an
element b ∈ V such that the values of b at x1 , x2 , . . . , xd are distinct.
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
35
Proof. For a vector space V0 ⊆ V we let B0 ⊆ B be the sub-algebra generated
by V0 . There is a finite dimensional vector space V0 ⊆ V such that the images
of x1 , x2 , . . . , xd in Spec(B0 ⊗k k 0 ) are distinct. Replacing V and B with V0 and
B0 we can thus assume that V is finite dimensional. It is further clear that we
can assume that B = S(V ). The points x1 , x2 , . . . , xd then corresponds to vectors
of V ∨ ⊗k k 0 and we need to find a k-rational hyperplane which does not contain
the d2 difference vectors xi − xj . A similar counting argument as in the proof of
Lemma (3.1.6) shows that if k has at least d2 elements then this is possible.
Proposition (5.2.11). Let A be a ring and B an A-algebra. Let V ⊂ B be an
A-submodule such that B is generated by V as an algebra. Consider the following
three ideals of ΓdA (B)
(i) The canonical ideal I1 = δ(x, y) x,y∈B d .
(ii) I2 = δ(x, x) x∈B d .
(iii) I3 = δ(x, x) x=(1,b,b2 ,...,bd−1 ), b∈V .
The closed subsets determined
by I1 and
I2 coincide with the degeneracy locus of
Γd Spec(B)/Spec(A) = Spec ΓdA (B) . If every residue field of A has at least d2
elements then so does the closed subset determined by I3 .
Proof. The discussion in (5.2.3) shows that it is enough to prove that the image of
d
d
the ideals Ik by the homomorphism ΓdA (B)
→ TSA (B) ,→ TA (B) set-theoretically
d
defines the big diagonal of Spec TA (B) . By Proposition (5.2.7) the image of
δ(x, y) is a(x)a(y).
Thus the radicals of the images of I1 and I2 equals the radical
of J = a(x) x∈B d . It is further easily seen that J is contained in the ideal of
every diagonal of Spec TdA (B) . Equivalently, the closed subset corresponding to
J contains the big diagonal.
By Lemmas (5.2.8) and (5.2.9) it is enough to show the first part of the proposition after any base change A → A0 such that Spec(A0 ) → Spec(A)
is surjective. We
can thus assume that every residue field of A has at least d2 elements. Both parts
of the proposition then follows if we show that the closed subset corresponding to
the ideal
K = a(1, b, b2 , . . . , bd−1 ) b∈V ⊆ TdA (B)
is contained in the big diagonal. As the formation of the ideal K commutes with
base changes A → A0 which are either surjections or localizations we can replace A
with one of its residue fields and assume
that A is a field with at least d2 elements.
d
Let Spec(k) : x → Spec TA (B) be a point corresponding to d distinct k-points
x1 , x2 , . . . , xd of Spec(B ⊗A k). Lemma (5.2.10) shows that there is an element
b ∈ V which takes d distinct values a1 , a2 , . . . , ad ∈ k on the d points. The value of
36
DAVID RYDH
a(1, b, b2 , . . . , bd−1 ) at x is then
1
1

= det  .
 ..
a1
a2
..
.
a21
a22
..
.
...
...
..
.

a1d−1
a2d−1 
 Y
(ai −aj )
..  =
. 
1
ad
a2d
...
add−1

X
σ∈Sd
σ(1)−1 σ(2)−1
a2
(−1)|σ| a1
σ(d)−1
. . . ad
j<i
which is non-zero. Thus x is not contained in the zero-set of K. This shows that
zero-set of K is contained in the big diagonal and hence that zero-set defined by K
is the big diagonal.
5.3. Non-degenerated symmetric tensors and divided powers.
Vd
Lemma (5.3.1). Let A be a ring, B an A-algebra and x, y ∈ A (B). Then
ΓdA (B)δ(x,y) → TSdA (B)δ(x,y) is an isomorphism.
Proof. Denote the canonical homomorphism ΓdA (B) → TSdA (B) with ϕ. Let f ∈
TSdA (B). As the anti-symmetrization operator a : TdA (B) → TdA (B) is a TSdA (B)module homomorphism we have that f a(x) = a(f x). By Proposition (5.2.7)
f ϕ δ(x, y) = f a(x)a(y) = a(f x)a(y) = ϕ δ(f x, y)
which shows that ϕ is surjective after localizing in δ(x, y).
To show that ϕδ(x,y) is injective, it is enough to show that the composition
ΓdA (B)δ(x,y) → TSdA (B)δ(x,y) ,→ TdA (B)δ(x,y)
is injective. Choose a surjection F B with F a flat A-algebra and let I be the
kernel of F B. Let J be the kernel of TdA (F ) TdA (B).
Let f ∈ J G . As f ∈ J we can write f as a sum f1 + f2 + · · · + fn such that for
every i we have that fi = fi1 ⊗ fi2 ⊗ · · · ⊗ fid ∈ TdA (F ) with fij ∈ I for some j.
Vd
Vd
Choose liftings x0 , y 0 ∈ A (F ) of x, y ∈ A (B). Identifying ΓdA (F ) and TSdA (F ),
we have that f δ(x0 , y 0 ) = δ(f x0 , y 0 ). This is a sum of determinants with elements
in ΓdA (F ) such that in every determinant there is a row in which every element is
in the ideal γ 1 (I) × γ d−1 (1). Thus δ(f x0 , y 0 ) is in the kernel of ΓdA (F ) ΓdA (B)
by (1.1.8). The image of f in ΓdA (B) is thus zero after multiplying with δ(x, y).
Consequently ϕ is injective after localizing in δ(x, y).
Theorem (5.3.2). Let X/S be a separated algebraic space. Then Symd (X/S)nd →
Γd (X/S)nd is an isomorphism.
Proof. We can assume that S and X are affine (3.2.5). The theorem then follows
from Proposition (5.2.11) and Lemma (5.3.1).
Definition (5.3.3). Let A be any ring and B = A[x1 , x2 , . . . , xr ]. We call the
elements f ∈ ΓdA (B) of degree one, see Definition (1.3.2), multilinear or elementary
multisymmetric functions. These are elements of the form
γ d1 (x1 ) × γ d2 (x2 ) × · · · × γ dn (xn ) × γ d−d1 −···−dn (1).
HILBERT AND CHOW SCHEMES, SYM. PRODUCTS AND DIV. POWERS
37
We let ΓdA (A[x1 , x2 , . . . , xn ])mult.lin. denote the subalgebra of ΓdA (A[x1 , x2 , . . . , xn ])
generated by multi-linear elements.
Remark (5.3.4). If the characteristic of A is zero or more generally if d! is invertible
in A, then ΓdA (A[x1 , x2 , . . . , xn ])mult.lin. = ΓdA (A[x1 , x2 , . . . , xn ]) by Theorem (1.3.4).
Lemma (5.3.5). Let A be a ring and B = A[x1 , x2 , . . . , xn ]. Let b ∈ B1 and let x =
(1, b, b2 , . . . , bd−1 ). Then ΓdA (B)mult.lin. δ(x,x) ,→ ΓdA (B)δ(x,x) is an isomorphism.
Proof. Let f ∈ ΓdA (B) = TSdA (B). We will show that f is a sum of products
of multilinear elements after multiplication by a power of δ(x, x). As f δ(x, x) =
δ(f x, x) and the latter is a sum of products of elements of the type γ 1 (c) × γ d−1 (1)
we can assume that f is of this type. As c 7→ γ 1 (c) × γ d−1 (1) is linear we can
further assume that c = xα for some non-trivial monomial xα ∈ B. It will be useful
to instead assume that c = xα bk with |α| ≥ 1 and k ∈ N. We will now proceed on
induction on |α|.
Assume that |α| = 1. If k = 0 then f = γ 1 (xα bk ) × γ d−1 (1) is multilinear. We
continue with induction on k to show that f ∈ ΓdA (B)mult.lin. . We have that
f = γ 1 (xα bk ) × γ d−1 (1) = γ 1 (xα bk−1 ) × γ d−1 (1) γ 1 (b) × γ d−1 (1)
− γ 1 (xα bk−1 ) × γ 1 (b) × γ d−2 (1)
and by induction it is enough to show that the last term is in ΓdA (B)mult.lin. . Similar
use of the relation
γ 1 (xα bk−` )×γ ` (b)×γ d−`−1 (1) = γ 1 (xα bk−`−1 )×γ d−1 (1) γ `+1 (b)×γ d−`−1 (1)
− γ 1 (xα bk−`−1 ) × γ `+1 (b) × γ d−`−2 (1)
with 1 ≤ l ≤ d − 2 and l ≤ k − 1 shows that it is enough to consider either
γ 1 (xα ) × γ k (b) × γ d−k−1 (1) if k ≤ d − 1 or γ 1 (xα bk−d+1 ) × γ d−1 (b) if k > d −
1. The first element of these is multilinear and the second is the product of the
multilinear element γ d (b) and γ 1 (xα bk−d ) × γ d−1 (1) which by the induction on k is
in ΓdA (B)mult.lin. .
0
00
If |α| > 1 then xα = xα xα for some α0 , α00 such that |α0 |, |α00 | < |α|. We have
that
0
00
f = γ 1 (c) × γ d−1 (1) = γ 1 (xα bk ) × γ d−1 (1) γ 1 (xα ) × γ d−1 (1)
0
00
− γ 1 (xα bk ) × γ 1 (xα ) × γ d−2 (1).
By induction it is enough to show that the last term is a sum of products of
0
multilinear elements, after suitable multiplication by δ(x, x). Let g = γ 1 (xα bk ) ×
00
γ 1 (xα )×γ d−2 (1). Then gδ(x, x) = δ(gx, x) which is a sum of products of elements
0
0
00
00
d−1
of the kind γ 1 (xα bt )×γ
(1) and γ 1 (xα bt )×γ d−1 (1). By induction on |α| these
are in ΓdA (B)mult.lin. δ(x,x) .
38
DAVID RYDH
Theorem (5.3.6). Let X/S be quasi-projective morphism of schemes and let X ,→
PS (E) be an immersion for some quasi-coherent
OS -module E of finite type. Then
Γd (X/S)nd → Chow0,d X ,→ P(E) nd is an isomorphism.
Proof. As Γ commutes with arbitrary base change and Chow commutes with flat
base change we may assume that S is affine and, using Lemma (1.2.3), that every
residue field of S has at least d2 elements. If E 0 E is
a surjection of OS modules then Chow0,d X ,→ P(E) = Chow0,d X ,→ P(E 0 ) by Definition (3.3.8)
and we may thus assume that E is free. Further
as Chow0,d X ,→ P(E) is the
d
d
schematic image of Γ (X/S) ,→ Γ P(E)/S → Chow0,d P(E) we may assume
that X = P(E) = Pn .
By Proposition (3.3.10)and the assumption on the residue fields of S = Spec(A),
the scheme Chow0,d P(E) is covered
by affine open subsets over which the morphism Γd P(E)/S → Chow0,d P(E) corresponds to the inclusion of rings
ΓdA (A[x1 , x2 , . . . , xn ])mult.lin. ,→ ΓdA (A[x1 , x2 , . . . , xn ]).
The theorem now follows from Proposition (5.2.11) and Lemma (5.3.5).
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Department of Mathematics, KTH, 100 44 Stockholm, Sweden
E-mail address: [email protected]